On Quantum Effects of Vector Potentials and Generalizations of Functional Analysis

Chapman University Chapman University Digital Commons

Computational and Data Sciences (PhD) Dissertations Dissertations and Theses

Spring 5-2020

Ismael L. Paiva Chapman University, [email protected]

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Recommended Citation I. Paiva, "On quantum effects of vector potentials and generalizations of functional analysis," Ph.D. dissertation, Chapman University, Orange, CA, Year. https://doi.org/10.36837/chapman.000159

This Dissertation is brought to you for free and open access by the Dissertations and Theses at Chapman University Digital Commons. It has been accepted for inclusion in Computational and Data Sciences (PhD) Dissertations by an authorized administrator of Chapman University Digital Commons. For more information, please contact [email protected]. On Quantum Eﬀects of Vector Potentials and Generalizations of Functional Analysis

A Dissertation by

Ismael Lucas de Paiva

Chapman University

Orange, CA

Schmid College of Science and Technology

Submitted in partial fulﬁllment of the requirements for the degree of

Doctor of Philosophy in Computational and Data Sciences

May 2020

Committee in charge:

Jeﬀ Tollaksen, Ph.D., Chair

Yakir Aharonov, Ph.D.

Daniel Alpay, Ph.D.

Justin Dressel, Ph.D.

Augusto C. Lobo, Ph.D.

Sandu Popescu, Ph.D.

Daniele C. Struppa, Ph.D. pprov d.

Dani 1 Alpa Ph.D. IL_/ Ju in Dr 1, Ph.D.

ugusto C. Lobo Ph.D.

Sandu Pope cu Ph.D.

Daniel . Struppa Ph.D.

III ACKNOWLEDGMENTS

Many people and organizations contributed directly or indirectly to my studies. I wish to pay gratitude to them. First, I would like to thank my parents, Geraldo and Nádia,for their continuous and unconditional support throughout different stages of my life. Also, I would like to thank the friendship from my siblings, Rafael and Miriã.I appreciate the privilege of growing up knowing that my family would always be there for me.

I would also like to thank all teachers and professors that contributed to my academic formation. I am truly grateful to all of them. To mention a few (in chronological order of contribution to my studies), I should begin by giving a special thanks to KátiaVilela. She was one of my high school teachers and, besides all the effort put in her lectures, she also recommended me to participate in a Junior Scientific Initiation at the Federal University of Ouro Preto. It was then, studying with FlávioCassino, that I started making my mind about pursuing a career in physics. Following that, I was advised by Augusto Lobo during my undergraduate. I have no words good enough to thank him. His advising continued during my masters, this time with the participation of Maria Carolina Nemes, who is not with us anymore but had a big impact on me with her constant excitement about physics and her dedication to her students.

Finished my masters, I started my applications to start my doctorate studies at Chapman University. If it were not the assistance from Jeﬀ Tollaksen, one of my Ph.D. advisors, since the time I was still in Brazil trying to get a fellowship, I would not have gotten here. He has supported me throughout this entire time, always trying to make sure I was taken care of. Moreover, I am incommensurately grateful for all the generosity of Yakir Aharonov. I do not know how to thank him for his help on my applications for the fellowship to come to Chapman, his eﬀort to guide me on my physics research, and his care. Furthermore, I would like to thank Daniel Alpay, who has advised me on my mathematical research. We started

IV working during a period where I was thinking that I would not succeed in my doctorate studies. Thank you for all the time you dedicated to working with me. I also thank Daniele Struppa for the collaboration in our projects. Finally, I am in great debit with Mordecai (Cai) Waegell. I do not believe I would have successfully concluded the physics research that is presented in this dissertation if it was not our long debates and for his important contributions. Thank you very much!

Besides the aforementioned names, I would like to also thank the physics professors at Chapman University. In particular, I thank Justin Dressel for our many discussions in his oﬃce. I learned a great deal with you. I am also grateful for the discussions and interactions with Matthew Leifer, Roman Buniy, and Amir Ahsan.

I would also like to thank all my friends. In special, I thank Shiva Lotfallahzadeh Barzili. From getting at Chapman as international students up until now, we shared our journey through the doctorate studies. Thank you for your friendship. I am also grateful for my landlords, Lawrence and Lois Reed. They gave me support since my ﬁrst weeks in the United States. They also inserted me in their family in such a way that, now, they are a second family to me. Besides them, I would also like to mention some of the friends who were important in my journey (in alphabetic order): Kyle Anderson, Lorenzo Catani, Dana Dacier, Sidy Danioko, Erik de Oliveira Martins, Pedro Dieguez, Cristhiano Duarte, Sam Ford, Eric Freda, Luis Pedro Garc´ıa-Pintos, Razieh Mohseninia, Robin Pendergraft, Cyril Rakovski, Viseth Sean, Arjun Tummala, Adrian Vajiac, Mihaela Vajiac, Kristen Whitney, Jianwei (Arnold) Zheng. This list of names is not intended to be complete, and I ask pardon from anyone who was not included in it.

Moreover, I would like to thank all the funding agencies/institutions that supported me. In special, I would like to thank the Science without Borders program from the Brazilian government. Unfortunately, it does not exist anymore, but I would never have dreamed of

V studying in a different country if it was not for this program. The fellowship that I got through this program was funded by the National Council for Scientific and Technological Development (CNPq), an agency that has supported me in different stages of my studies. I am grateful that I was studying at a time when so many opportunities were available to students like me. I also thank Chapman University for the teaching assistantship that was offered to me. Furthermore, I thank the Fetzer Franklin Fund of the John E. Fetzer Memorial Trust, which provided some of the funds that were used in travels related to my research.

Finally, I thank the members of the committee for the time and work put on reading this dissertation. In special, I would like to thank Sandu Popescu, who was not mentioned by name in these acknowledgments yet. I really value the participation of every committee member and their suggestions for future works.

To all mentioned here, muit´ıssimoobrigado!

VI LIST OF PUBLICATIONS

D. Alpay, I. L. Paiva, and D. C. Struppa, Distribution spaces and a new construction of stochastic processes associated with the Grassmann algebra, J. Math. Phys. 60, 013508 (2019).

I. L. Paiva, Y. Aharonov, J. Tollaksen, and M. Waegell, Topological bound states for quantum charges, Phys. Rev. A 100, 040101(R) (2019).

D. Alpay, I. L. Paiva, and D. C. Struppa, Positivity, rational Schur functions, Blaschke factors, and other related results in the Grassmann algebra, Integr. Equ. Op. Th. 91, 8 (2019).

I. L. Paiva, Y. Aharonov, J. Tollaksen, and M. Waegell, Magnetic forces in the absence of a classical magnetic ﬁeld, Phys. Rev. A 101, 042111 (2020).

D. Alpay, I. L. Paiva, and D. C. Struppa, A general setting for functions of Fueter variables: diﬀerentiability, rational functions, Fock module and related topics, Isr. J. Math. 236, 207 (2020).

D. Alpay and I. L. Paiva, On the extension of positive deﬁnite kernels to topological algebras, arXiv:2003.05632 (2020).

I. L. Paiva, Y. Aharonov, J. Tollaksen, and M. Waegell, Complex-valued vector potential in a pre and post-selected system, (in preparation) (2020).

VII ABSTRACT

On Quantum Eﬀects of Vector Potentials and Generalizations of Functional Analysis by Ismael Lucas de Paiva

This is a dissertation in two parts. In the first one, the Aharonov-Bohm effect is investigated. It is shown that solenoids (or flux lines) can be seen as barriers for quantum charges. In particular, a charge can be trapped in a sector of a long cavity by two flux lines. Also, grids of flux lines can approximate the force associated with continuous two-dimensional distributions of magnetic fields. More, if it is assumed that the lines can be as close to each other as desirable, it is explained how the classical magnetic force can emerge from the Aharonov- Bohm effect. Continuing, the quantization of the source of the magnetic field, and not just of the degrees of freedom of the particle interacting with it, is considered. Special attention is given to the cases where the source has a relatively small spreading and is post-selected. As it will be discussed, in those cases, the weak value plays a role in the determination of the effective vector potential “experienced” by the particle. In the second part of this work, notions from functional analysis are extended to Banach algebras and completions of Grassmann algebras. A notion of analyticity is given to the functions of a single Banach algebra variable. This notion allows the introduction of holomorphic polynomials, power series, and rational functions. With that, the analogous of Hilbert spaces of power series are also considered. Finally, closures of Grassmann algebras with respect to the 1 and the 2-norms are explored. The analogous of the complex analysis in the open disk or a half-plane (usually referred to as Schur analysis) is presented in the 1-norm closure. Also, a Wiener-like algebra, interpolation problems, and a process known as the Schur algorithm are studied in this setting. Now, an inner product between two elements can be introduced in the 2-norm closure, revealing similarities between this space and the non-commutative Fock-Bargamann- Segal space. It is, then, defined a class of stochastic processes. To conclude, the derivatives of these processes are analyzed in an analogous of the space of stochastic distributions.

VIII TABLE OF CONTENTS

ACKNOWLEDGMENTS ...... IV

LIST OF PUBLICATIONS ...... VII

ABSTRACT ...... VIII

LIST OF FIGURES ...... XII

LIST OF ABBREVIATIONS ...... XIII

LIST OF SYMBOLS ...... XIV

1 INTRODUCTION ...... 1

2 SOME LESSONS FROM THE AHARONOV-BOHM EFFECT ...... 8 2.1 Preliminaries ...... 8 2.2 Aharonov-Bohm Eﬀect ...... 14 2.3 Modular Variables ...... 17 2.4 Instantaneous Change in the Transverse Velocity ...... 21 2.5 Nodal Lines ...... 25 2.6 Equivalence Principle ...... 29

3 MAGNETIC FORCES FROM THE AHARONOV-BOHM EFFECT ...... 32 3.1 Continuous Magnetic Fields in Classical Physics ...... 32 3.2 Continuous magnetic ﬁelds in quantum mechanics ...... 34 3.3 Emulating Continuous Magnetic Fields with Distributions of Flux Lines . . . 37 3.4 Topological Bound States ...... 38 3.5 Landau Levels ...... 44 3.6 Semi-Classical Toy Model ...... 46 3.7 Discussion ...... 48

IX 4 COMPLEX VECTOR POTENTIALS IN PRE AND POST-SELECTED SYSTEMS 51 4.1 Weak Measurements and Weak Values ...... 51 4.2 Quantization of the Sources of Electromagnetic Fields ...... 59 4.3 Complex Vector Potentials ...... 63 4.4 Discussion ...... 67

5 FUETER VARIABLES ON BANACH ALGEBRAS ...... 69 5.1 Real Derivatives and Their Extension to Other Algebras ...... 69 5.2 The Algebra ...... 72 5.3 A General Principle ...... 75 m+1 5.4 Hyperholomorphicity of Functions from K into A ...... 79 5.5 Fueter Polynomials ...... 82 5.6 Fueter Series and the Gleason Problem ...... 85 5.7 Hyperholomorphic Rational Functions ...... 92 5.8 Banach Modules of Fueter Series ...... 95 5.8.1 Fock-Bargmann-Segal Module ...... 102 5.8.2 Drury-Arveson Module ...... 104 5.9 Discussion ...... 108

6 ANALYSIS IN GENERALIZED GRASSMANN ALGEBRAS ...... 110 6.1 Grassmann Algebra ...... 111 6.2 1-Norm Completion of the Grassmann Algebra ...... 117 6.3 Matrix Algebra and Extension of Toeplitz Matrices ...... 120 6.4 Realization Theory and Rational Functions ...... 126 6.5 Rational Schur-Grassmann functions ...... 131 6.6 Wiener-Grassmann algebra ...... 135 6.7 Reproducing Kernel Banach Modules and Interpolation ...... 140 6.8 Nevanlinna-Pick Interpolation ...... 145

X 6.9 Schur Algorithm ...... 148 6.10 Fock-Bargmann-Segal Space ...... 151

6.11 Topological Algebra Associated with Λ(2) ...... 155 6.12 Stochastic Processes and Their Derivatives ...... 164 6.13 Discussion ...... 172

REFERENCES ...... 176

XI LIST OF FIGURES

2.1 Illustration of an experiment that verifies the Aharonov-Bohm effect ...... 15 2.2 Interference experiment with a lattice of flux lines ...... 19 2.3 Real ground state of a circular cavity with a flux line carrying a semifluxon on its center ...... 28 2.4 Representation of a laboratory given by a narrow ring ...... 30 3.1 Classical particles traveling towards a region with magnetic field ...... 33 3.2 Representation of particles inside a cavity ...... 39 3.3 Schematic representation of a semi-classical theory where flux lines replace continuous magnetic fields ...... 47 4.1 Effect of measurements in the measurement device ...... 55 4.2 Effect of post-selections in weak measurements ...... 58 6.1 Schematic representation of the increasing family of Grassmann algebras . . . . 112 6.2 Schematic representation of a decreasing family of Hilbert spaces with increasing norm...... 156

XII LIST OF ABBREVIATIONS

AB Aharonov-Bohm

BA Banach algebra

BMFS Banach module of Fueter series

CK Cauchy-Kovalevskaya

DA Drury-Arveson

FBS Fock-Bargmann-Segal

FV Fueter variables

GA Grassmann algebra

HS Hilbert space

NP Nevanlinna-Pick

RF Rational function

SG Schur-Grassmann

WG Wiener-Grassmann

WV Weak value

XIII LIST OF SYMBOLS

N Natural numbers

N0 Union of N with {0}

Z Integers

Zn Integers module n

∗ Zn Set subtraction of Zn by {0}

R Real numbers

C Complex numbers

A Banach algebra

Λn Grassmann algebra with n generators

Λ Union of Λn for every n ∈ N

Λ(p) Closure of Λ with respect to the p-norm

|·i Dirac ket h·| Dirac bra h·|·i Inner product between a bra and a ket

Tr(·) Trace functional

Trs Partial trace over a subspace s h·i Expected value h·iw Weak value

XIV 1 Introduction

This dissertation is divided into two parts. In the ﬁrst part, it is studied some results involving a quantum particle in the presence of a magnetic ﬁeld, but never in direct contact with it.

A better comprehension of electromagnetism was an essential factor in the revolution that started in physics during the past century. This revolution gave rise to two theories that completely shifted the understanding of the foundation of the physical world: the theory of relativity and the theory of quantum mechanics.

The former emerged from the studies of the apparent necessity of an absolute reference frame in the study of electromagnetism. While the equations of motion in Newtonian mechanics are invariant under Galilean transformation, Maxwell’s equations, which are the basis of electromagnetism, are not. It was, then, conjectured that there existed a medium called ether, which was believed to be everywhere in the universe and would serve as an absolute system of reference where Maxwell’s equations assumed their well-known forms. However, the Michelson-Morley experiment, which was an interference experiment meant to detect diﬀerences in the speed of light in systems moving with respect to the ether, did not return the expected results. It seemed, some thought, that the ether was an elastic solid. Lord Kelvin, for instance, thought that “we may simply deny the scholastic axiom that two portions of matter cannot jointly occupy the same space, and may assert, as an admissible hypothesis, that ether does occupy the same space as ponderable matter, and that ether is not displaced by ponderable bodies moving through space occupied by ether” [1].

In parallel with that, Voight [2], in 1887, and Lorentz [3], in 1904, considered a transformation that consisted in a way to change the system of coordinates for which Maxwell’s equations are invariant. There is a very radical idea that is implied by this transformation: the speed of light is the same in every inertial system of reference. Moreover, because of that, the

1 coordinates of space and time are always mixed in a change of referential. Even though these ideas were, in a sense, already present in the works of Voight and Lorentz, they only got their due attention with one of the two seminal publications in 1905 by Albert Einstein [4], where he went straight to the crux of the problem. This new theory led to necessary corrections to the Newtonian mechanics since the Galilean transform had to be replaced by the new transform introduced by Voight and Lorentz, named Lorentz transform by Poincar´e [5], who likely was unaware of Voight’s work.

Quantum theory, on the other hand, can be traced back to 1900, when Plank, continuing his studies on the black body radiation, was led to the quantization of the energy exchange between a cavity and an electromagnetic ﬁeld within it [6]. In 1905, in his other seminal article published that year, Einstein introduced the idea that electromagnetic radiation is always quantized to explain the photoelectric eﬀect [7]. Inspired by these results, Bohr presented, in 1913, his model for a hydrogen atom [8], in which electrons had discrete levels of energy.

Later, in 1925, Heisenberg published a seminal article where he proposed a theory where the coordinates of position and the momentum of electrons in atoms were non-commuting physical quantities [9]. Following that, Dirac developed an algebraic formulation that provided a formal framework for the commutativity law [10]. Also, Born and Jordan further developed this theory in Ref. [11] and in a follow up article with Heisenberg [12]. The celebrated Heisenberg commutation relation was published in 1927 [13].

Moreover, in 1926, Pauli used Heisenberg’s matrix mechanics to compute the spectrum and the Stark eﬀect for a hydrogen atom [14]. However, the computation of the spectrum of more generic systems was overly complicated in this theory — at that time, physicists were not familiar with matrix algebra.

In parallel with Heisenberg’s theory, a wave mechanics theory was also being developed.

2 In 1923, de Broglie used the idea that electromagnetic waves were quantized, together with Eintein’s formula E = mc2, which was also published in 1905 [15], to suggest that corpuscular objects, i.e., masses, should also have a wavelength associated with them [16]. Motivated by it, in 1926, Schrödingerdeveloped his acclaimed equation to describe the dynamics of such waves [17], now called wave functions. Also, still in 1926, Born introduced the probabilistic interpretation for the wave function [12]. The theory emerging from these ideas was broadly diffused in the physics community since it required a mathematical knowledge that was more familiar to physicists: linear partial differential equations.

Finally, Schr¨odingerhimself noticed that matrix and wave mechanics were mathematically equivalent [18]. His work was followed by further analysis from Dirac [19, 20], Jordan [21–23], and von Neumann [24].

It would still take some time until the first discussions about a quantum charge having its dynamics affected by a magnetic field in a region it never enters. In a talk to a physical society meeting in Danzig in 1939, Franz discussed that the interference of an electron traveling in a region with magnetic depends on the total magnetic flux enclosed by it [25]. However, it is not clear if it was discussed the possibility of having the electron still being affected by the field without directly interacting with it [26].

Later, in 1949, Ehrenberg and Siday, in the context of the theory of refraction of electron rays, concluded that a magnetic flux, even if isolated, would affect the dynamics of a charge encircling it [27]. Their result went unnoticed — perhaps because it did not seem a general effect. In fact, their practical problem was the development of a magnetic lens for a beta spectrometer, and they concluded that “[o]ne might therefore expect wave-optical phenomena to arise which are due to the presence of a magnetic field but not due to the magnetic field itself, i.e., which arise whilst the rays are in field-free regions of space.” Finally, unaware of the Ehrenberg and Siday’s article, Aharonov and Bohm published their seminal work in

3 1959 [28]. Besides also introducing an electric version of it, they went straight to the crux of the problem, making the fundamental character of the result evident. With that, their work got traction since its publication. The effect, which is known as the Aharonov-Bohm (AB) effect, was observed experimentally [29–32], and it has been the subject of a vast study in the literature [32–53]. Also, a version of the effect for neutral particles with a magnetic moment, usually referred to as the Aharonov-Casher effect, was introduced [54]. Moreover, the effect still suscitates debates about its non-local nature [55–65].

Here, new consequences of the AB effect are presented. One of them, which can also be found in Refs. [66, 67], shows that solenoids (or flux lines) can be seen as barriers for quantum charges. In particular, two flux lines can trap a particle in a sector of a long cavity. Moreover, grids of flux lines can approximate the force associated with continuous two-dimensional distributions of magnetic fields. More, assuming the lines can be as close to each other as desirable, it is explained how the classical magnetic force can emerge from the AB effect.

Furthermore, as can also be seen in Ref. [68], the effect is considered in the presence of a quantized source of magnetic fields. When the source is post-selected, the weak value, as it will be studied, plays a role in the determination of the effective vector potential “experienced” by the particle.

In the second part of this dissertation, a variety of results from functional analysis, operator theory, and the theory of stochastic distributions is extended to a Banach algebra (BA) with an involution and completions of Grassmann algebras (GAs). A structure underlying most of this study is the concept of a Hilbert space (HS) of power series. HSs appeared as generalizations of ﬁnite-dimensional linear spaces. They were the result of a shift in the view of sequences — and, later, of functions — that arose in the study of linear integral equations. Instead of being seen as isolate objects, they started being analyzed as elements

4 of a space. Part of the origin of HSs can be associated with Hilbert’s study of linear integral equations in a series of six papers published by him between 1904 and 1910, and composed his acclaimed book from 1912 [69].

In parallel to his work, Lebesgue introduced his notorious integral in 1904 [70]. The connection with HSs appeared in 1907 with Schmidt and Fréchet showing that the space of square summable functions, which is associated with a particular type of Lebesgue integration, had a geometry similar to HSs. Following that, Riesz showed that, in fact, there exists a one-to- one correspondence between HSs of sequences and Lebesgue square measurable functions, which allowed Fréchet and Fisher to show that these functions form a complete metric space. More, Fréchet introduced an integral formula for the action of continuous linear functional on elements of these metric spaces [71]. Also, Fréchet [72] and Riesz [73] introduced important properties of the dual of HSs with what is now known as the Riesz representation theorem and, in 1908, Schmidt proved a type of spectral theorem [74]. Moreover, existing results by Bessel, Fourier, and Parseval were translated into this new framework.

The ﬁrst complete formal treatment of HSs was presented by von Neumann. He, then, together with Hilbert and Nordheim, applied this formalism to quantum mechanics in 1928 [75]. After that, other applications in physics (including in classical mechanics) and in other areas (like signal processing) were found [76, 77]. Finally, it should also be mentioned that the name Hilbert space was only introduced by von Neumann in 1930 [78].

The HSs of interest here are HSs of power series. In particular, it is studied the Hardy space, formally introduced by Riesz in 1923 [79], but named by him after Hardy because, in 1915, Hardy had worked on the functions he was considering [80]. Another HS of power series of interest is the Fock-Bargmann-Segal space, which was introduced by Fock in 1932 [81] as the completion of a direct sum of HSs in the study of identical quantum particles. Later, Bargmann [82], in 1961, and Segal [83], in 1963, presented a HS of power series isomorphic

5 to the Fock space. Finally, a space introduced by Drury [84], which had its importance highlighted by Arveson [85], often referred to as the Drury-Arveson space, is also considered in this work.

These and other HSs of power series were extended from a single complex variable to more general settings, like several complex variables, upper triangular operators, quaternions, and bi-complex numbers. They have been a source of new problems and methods [86–91]. Here, as already mentioned, some results on the extension of these spaces to BAs and completions of GAs are presented.

In the extension to BAs, which is a work originally presented in Ref. [92], a notion of analyticity is given to functions of a single variable in this setting, which, later, allows the introduction of the analogous of HSs of power series. Now, in the case of GAs, their closure with respect to the 1 and the 2-norms are considered. The 1-norm closure was studied in Ref. [93]. The convergence of power series and interpolation problems are investigated in this setting. The space of functions of a single complex variable with image in the closure of the Grassmann algebra with respect to the 2-closure, however, presents a structure similar to the non-commutative FBS space. A class of stochastic processes, which includes the analogous of the fractional Brownian motion, is deﬁned and studied here, as done in Ref. [94].

This work is structured as follows. Chapter 2 brieﬂy introduces some basic concepts from quantum mechanics and electromagnetism. This provides some tools for the presentation of the AB eﬀect. Also, some results associated with it are reviewed.

Chapter 3 discusses a peculiar consequence of the AB effect: flux lines introduce a field- free force on quantum charges that can emulate continuous two-dimensional magnetic fields. This result is explored, leading to the obtainment of topological bound states and to the approximation of Landau levels. It is also discussed a semi-classical theory where grids of flux lines can be seen as the source of the classical magnetic force, at least for two-dimensional

6 ﬁelds.

Chapter 4 consider a case where the AB effect happens in the presence of quantized sources of magnetic fields. For that, a study on the quantization of sources is presented. Weak interactions between the source and the charge are of particular interest, and, because of it, the concepts of weak measurements and weak values are reviewed. Finally, it is shown how the effective vector potential “experienced” by the particle is complex-valued.

Chapter 5 starts the mathematical explorations of this work. A notion of derivative analogous to the one presented by Fueter for functions of a single quaternionic variable is extended to the case where the variable takes value in a BA that satisﬁes a minimum set of properties. With this result, holomorphic polynomials, power series, and rational functions are introduced. This, in turn, allows the study of Banach modules of holomorphic power series, the counterpart of HSs of power series in this context.

Finally, Chapter 6 considers various analysis problems in completions of GAs with respect to the 1 and the 2-norm. The analogous of the complex analysis in the open disk or a half- plane, often referred to as Schur analysis, is presented in the 1-norm closure. In particular, a Wiener-like algebra, which plays a role of the analogous of the Hardy space, is considered. Also, interpolation problems and a process called Schur algorithm are studied in this setting. In the 2-norm closure, it is noted that the inner product between two elements can be introduced, revealing similarities between this space and the non-commutative FBS space. It is, then, deﬁned a class of stochastic processes. The derivatives of these processes are analyzed in a space analogous to the space of stochastic distributions.

It should be noted that, rather than including a chapter with ﬁnal remarks, it was chosen to incorporate a discussion section at the end of every chapter that introduces a result that is a product of the studies for this dissertation.

7 2 Some Lessons from the Aharonov-Bohm Eﬀect

As mentioned in the introduction, the Aharonov-Bohm (AB) effect is a vastly studied phenomenon, which has led to the development of many theoretical and experimental tools and numerous surprising consequences. This chapter presents an explanation of the effect and reviews some results that will be useful for the next two chapters. To start, a brief introduction to some basic concepts and definitions in quantum mechanics is given.

2.1 Preliminaries

A closed quantum system is represented by a vector, denoted by |ψi, in a complex Hilbert space (HS) H. Its adjoint is denoted by hψ| = (|ψi)†. Physical quantities that can be associated with the system and measured are given by self-adjoint operators acting on H. They are often referred to as observables.

The spectral theorem assures that there exists an orthogonal basis {|oi} such that a self- adjoint operator O can be written as a diagonal operator

Z O = o|oiho|do. (2.1)

The elements |oi are called the eigenvectors (or eigenstates) of O and o are the (real) eigenvalues associated with them

If H admits a discrete basis, Eq. (2.1) becomes a sum of the type

N−1 X O = ok|okihok|, (2.2) k=0

8 where N is the dimension of H, which can be infinite. Also, the norm of the vectors of interest is always one, i.e., hψ|ψi = 1, (2.3) where h·|·i denotes the inner product in H. The reason is that there exists a probabilistic interpretation of the state, given by the Born rule: if the observable O is measured and the state of the system is |ψi, the probability of finding the system in the state |oki is give by 2 |hok|ψi| . Observe that the absolute value is necessary because hok|ψi, called the amplitude of probability, is, in general, a complex number. Moreover, with the choice associated with Eq. (2.3), the quantum states are defined up to a global phase. Then, two normal vectors in a complex HS represent the same quantum state if they differ by a global phase. Also, note that

hoj|oki = δjk, (2.4) where δjk is the Kronecker delta. The change on the state that describes the system from P |ψi = k ψk|oki to |oki is what is often referred to as the collapse of the state caused by the measurement of O. Furthermore, the average over repeated measurements of an observable O over a system in the state |ψi results in the expected value hOi of O, which is given by

X X hOi = hψ|O|ψi = okψjψkhoj|oki = okψkψk. (2.5) j,k k

Similar results hold in the case where the HS has a continuous basis. However, the relation between two elements of a basis is

ho|o0i = δ(o − o0), (2.6)

9 where δ(o − o0) is the Dirac delta. Also, the expected value of hOi becomes

Z hOi = hψ|O|ψi = oψ(x)ψk(x)dx. (2.7)

The position (operator X) and the momentum (operator P ) of a quantum particle are given by a vector in a continuous HS. Then, if the position of a system is described by the state |ψi, the wave function ψ(x) = hx|ψi describes the probability amplitude of ﬁnding the particle at a certain position x. Observe that ψˆ(p) = hp|ψi can also be deﬁned. In fact, ψˆ(p) is the Fourier transform of ψ(x). Also, X and P satisfy the well-known canonical commutation relation

[X,P ] ≡ XP − PX = i~I, (2.8) where I is the identity operator.

The composition of quantum systems is given by the tensor product. Then, if a state |ψi belongs to H1 and a state |ϕi belongs to H2, their joint state is

|Θi = |ψi ⊗ |ϕi ∈ H1 ⊗ H2. (2.9)

A state that can be written in this form is called a product or a separable state. However, a

general state of H1 ⊗ H2 is written as

ZZ |Γi = γ(r, s)|ri ⊗ |si dr ds, (2.10)

where |ri denotes a basis of H1 and |si denotes a basis of H2. If a state |Γi cannot be written as a pure state, each of its parts is called an entangled or a non-separable state. In this case, the state associated with each system can be written with the deﬁnition of a partial trace,

10 i.e., the state of system 1 is a partial trace of |ΓihΓ| over the system 2

Similarly, the state of system 2 is

ZZZ 0 0 0 ρ2 ≡ Tr1|ΓihΓ| = γ(r, s )γ(r, s)|sihs | ds ds dr. (2.12)

Observe that, in this case, the states are no longer vectors in H1 and H2. They are operators, called density matrices, acting on those spaces. Although this is the most general form to represent a system, most of the time only vectors in HSs are considered here.

The dynamics of quantum systems is governed by Hamiltonians. The Schr¨odingerequation

states that a Hamiltonian H is the generator of translations in time of the a state |ψ(t0)i = |ψi, i.e., ∂ i |ψ(t)i = H|ψ(t)i. (2.13) ~∂t

This means that there exists an unitary U such that

∂ ∂ i U(t; t )|ψi = HU(t; t )|ψi ⇒ i U = HU. (2.14) ~∂t 0 0 ~∂t

In general, the Hamiltonian is a function of time and Eq. (2.14) has to be carefully evaluated. However, in cases where H(t) commutes with H(t0) for every t and t0, Eq. (2.14) leads to

R t 0 0 − H(t )dt /~ U(t; t0) = e to . (2.15)

11 A common type of Hamiltonian is

1 H = P 2 + V (X,~ t), (2.16) 2m where V is an scalar potential, the canonical momentum P~ is the “vector of operators”

(Px,Py,Pz), and X~ = (X,Y,Z). In this case, the Schr¨odingerequation for a state |ψ(t)i in the position basis is

∂ψ 2 i (~x,t) = − ~ ∇2ψ(~x,t) + V (~x,t)ψ(~x,t), (2.17) ~ ∂t 2m where it was used the fact that

P~ = −i~∇~ . (2.18)

Observe that, because systems are, in general, entangled, the dynamics of an individual system is not always unitary. Here, however, this matter is not looked into.

Besides the Schr¨odingerpicture, there is another way to consider dynamics in quantum mechanics, which is given by the Heisenberg picture. In this perspective, even though quantum states still identiﬁes a system, they do not evolve. What evolves are their properties, i.e., the observables. The dynamics of the observables are governed by the Heisenberg equation

d i ∂O O(t) = [H,O(t)] + (t). (2.19) dt ~ ∂t

Most of the systems considered here are in the presence of an electromagnetic ﬁeld. Because of it, instead of the Hamiltonian in Eq. (2.16), their dynamics are governed by

1 2 H = P − qA~(X,~ t) + qV (X,~ t) + V (X,~ t), (2.20) 2m q where A~ is the vector potential and Vq is the scalar potential. These potentials are such that

12 the electric ﬁeld is ∂A~ E~ = − − ∇~ V (2.21) ∂t q

and the magnetic ﬁeld is B~ = ∇~ × A.~ (2.22)

Because the potentials are defined as a way to mathematically “reconstruct” the fields E~ and B~ , it can be seen from Eqs. (2.21) and Eq. (2.22) that there is not a unique way to define

A~ and Vq. In fact, if Λ is a function of the position coordinates and time, the potentials

A~0 = A~ + ∇~ Λ (2.23)

and ∂Λ V 0 = V − (2.24) q q ∂t

are associated with the same electromagnetic ﬁeld. The change from the potentials A~ and ~0 0 Vq to A and Vq is called a gauge transformation. Also, Λ is associated with a gauge choice.

To conclude this section, it should be noted that, in quantum mechanics, the gauge invariance implies that, given the unitary ~ U = eiqΛ(X,t)/~, (2.25) the Hamiltonian in Eq. (2.20) is physically equivalent to the Hamiltonian

∂Λ H0 = UHU† − q (X,~ t), (2.26) ∂t

as can be checked by direct computation and observing that Eq. (2.8) implies that

h i P~ , f(X~ ) = i~∇~ Λ(X~ ). (2.27)

13 With that set, the AB eﬀect and its consequences can ﬁnally be studied.

2.2 Aharonov-Bohm Eﬀect

In classical physics, the dynamics of a particle with charge q is only affected by a magnetic field which it directly interacts with it, i.e., if it travels in a region where the magnetic field is non-zero. However, in quantum mechanics, this is not always the case. If a charge encircles a region in space that contains a magnetic field, it accumulates a quantum phase that is proportional to the magnetic flux inside the region enclosed by its trajectory, regardless of whether there was any magnetic field on the trajectory of the particle. This is the (magnetic) AB effect, which was introduced in 1959 by Aharonov and Bohm [28].

The way this effect is usually presented is by considering a charge encircling a solenoid whose axis lies in the z axis, as represented by Fig. 2.1. For simplicity, the solenoid is taken to be infinitely thin — in which case it will be referred to as a flux line. Also, the particle is assumed to travel in the xy plane on the superposition

1 |Ψi = √ (|ψLi + |ψRi) , (2.28) 2

where |ψLi is a packet that passes to the left and |ψRi is a packet that passes to the right of the ﬂux line. This superposition can be achieved, for instance, with the use of a double-slit. The Hamiltonian of the particle can be written as

1 ~ 2 H = P~ − qA~(X~ ) , (2.29) 2m where P~ = Pxxˆ + Pyyˆ and X~ = Xxˆ + Y yˆ.

0 0 To obtain the solution of this Hamiltonian, let the states |ψLi and |ψRi be the solutions of

14 Figure 2.1: Illustration of an experiment that verifies the AB effect. A charge travels in a superposition of a state |ψLi traveling to the left and a state |ψRi traveling to the right of the solenoid (blue object), as represented by the red paths. This causes a shift in the interference pattern proportional to the magnetic flux inside the solenoid.

the case where there is no magnetic ﬁeld in the ﬂux lines, i.e.,

1 H = P 2. (2.30) 0 2m

Then, following a procedure introduced by Dirac [95], it can be obtained that, after encircling the ﬂux line, R ~ iq γ A·d`/~ 0 |ψLi = e L |ψLi (2.31)

and R ~ iq γ A·d`/~ 0 |ψRi = e R |ψLi, (2.32)

where γL and γR are, respectively, the trajectory of the center of |ψLi and |ψRi. Then, because quantum states are equivalent up to a global phase,

Z Z I A~ · d` − A~ · d` = A~ · d`, (2.33) γR γL

15 and the magnetic ﬂux ΦB inside the region enclosed by the charge is

I ~ ΦB = A · d`, (2.34) the state of the system after it encircles the ﬂux line is

1 |Ψi = √ |ψ0 i + eiϕAB |ψ0 i , (2.35) 2 L R where qΦB ϕAB ≡ mod 2π (2.36) ~ is the quantum phase accumulated by the charge, usually called the AB phase. The state in Eq. (2.35) should be compared to

1 |Ψ0i = √ |ψ0 i + |ψ0 i , (2.37) 2 L R which is the final state of the system that undergoes a regular double-slit experiment. Then, if an interference pattern is obtained from a double-slit experiment without a flux line, this pattern will be shifted by an amount determined by the AB phase if the same experiment is performed with a flux line placed in between the slits.

Finally, observe that, since a phase has a periodicity of 2π, the AB phase contains just the modular part of qΦB/~. Thus, the magnetic ﬂux

2π Φ ≡ ~, (2.38) 0 q called a ﬂuxon, creates an equivalence relation for which two congruent magnetic ﬂuxes modulo Φ0 give rise to the same AB phase.

16 2.3 Modular Variables

Consider a particle in the superposition

1 iα |ψi = √ |ψ1i + e |ψ2i , (2.39) 2 where ψ1(x) ≡ hx|ψ1i and ψ2(x) ≡ hx|ψ2i are wave packets, i.e., wave functions localized in a certain interval, with the same proﬁle, meaning that, up to a shift in their domain, they are represented by the same function. The question that will be addressed now concerns the detection of the relative phase α between ψ1 and ψ2. It is know that this phase can be observed by interference experiments, i.e., the phase can be detected when there is an overlap between the two wave packets. Nevertheless, it can be asked whether there exists an observable capable of detecting α when ψ1 and ψ2 do not overlap.

In principle, it seems that such an operator does not exist. The reason is that it would have to be written as a linear combination of powers of the operators of position and momentum.

However, for every n, m ∈ N0,

Z hXnP mi = ψ(x)XnP mψ(x)dx (2.40) 1 Z Z = ψ (x)XnP mψ (x)dx + ψ (x)XnP mψ (x)dx , 2 1 1 2 2 where it was used the fact that, because ψ1 and ψ2 do not overlap,

Z Z iα n m −iα n m e ψ1(x)X P ψ2(x)dx = e ψ2(x)X P ψ1(x)dx = 0. (2.41)

n m Therefore, hX P i does not depend on α for every n, m ∈ N0.

However, suppose that the center of the packets are separated by a distance L, i.e., ψ1(x) = iP L/ ψ2(x − L) or ψ1(x) = ψ2(x + L). Also, consider the operator e ~, which is the operator

17 of translations by L — more precisely, for any function f that admits a Fourier transform,

1 Z eiP L/~f(x) = eiP L/~ √ fˆ(p)eipx/~dp 2π~ 1 Z = √ fˆ(p)eip(x+L)/~dp (2.42) 2π~ = f(x + L), where fˆ is the Fourier transform of f. Then,

Z heiP L/~i = ψ(x)eiP L/~dx 1 Z Z = eiα ψ (x)ψ (x + L)dx + e−iα ψ (x)ψ (x + L)dx (2.43) 2 1 2 2 1 1 = eα 2 if ψ1(x) = ψ2(x + L). Hence, this operator is capable of detecting the phase α. Moreover, while eiP L/~ is not Hermitian and, thus, it is not an observable, the operators

PL eiP L/~ + e−iP L/~ cos = (2.44) ~ 2 and PL eiP L/~ − e−iP L/~ sin = (2.45) ~ 2i are, and they can also detect α. In fact, hcos(P L/~)i = (cos α)/2 and hsin(P L/~)i = (sin α)/2. Then, it is possible, in principle, to construct observables capable of detecting the relative phase of two non-overlapping wave packets.

It should be noticed that only the value

2π P ≡ P mod ~ (2.46) mod L

......

Figure 2.2: Quantum charge q (yellow cloud) sent through a lattice of flux lines (blue dots), each carrying a magnetic flux ΦB. Low-energy particles are reflected. The gray lines are just horizontal lines that cross the lattice in the midpoint between two flux lines and are not part of the configuration of the system.

iP L/ is relevant for e ~ and any other operator derived from it. The operator Pmod is called the modular momentum of the charge and it is a type of modular variable — a concept introduced in 1969 by Aharonov, Pendleton and Petersen [96], and further developed after that [58, 97–100]. Modular variables of position and energy can be also deﬁned [58].

Before proceeding, observe that, from Eq. (2.46), it can be introduced a Hermitian operator N such that 2π P = ~N + P . (2.47) L mod

As an application of this concept, consider the scenario represented in Fig. 2.2 of a charge traveling towards a lattice, which coincides with the y axis, of impenetrable lines spaced by a length L. Then, the Hamiltonian of the particle can be written as

1 H = P 2 + P 2 + V (X,Y ), (2.48) 2m x y

19 where V is a potential associated with lattice. Observe that V (x, y) = 0 for every x 6= 0. Moreover, it is periodic in y when x = 0. Using the Heisenberg equation of motion, it can, then, be observed that, for x 6= 0,

d i Py = H,Py = 0, (2.49) dt ~

i.e., there is no change in the momentum Py. The same can be said about Px. Now, at x = 0, H,Py 6= 0, which implies that Py is, in general, not conserved. However,

d i h i ei(Py)modL/~ = H, eiPyL/~ dt ~ i h i = V (Y ), eiPyL/~ (2.50) ~ i = (V (Y ) − V (Y + L)) ei(Py)modL/~, ~

where V (y) ≡ V (0, y). Because of the periodicity of V (y), the term between parenthesis in the last part of Eq. (2.50) vanishes, i.e., there is no change in the modular momentum

(Py)mod. Thus, from Eq. (2.47), it can be concluded that the momentum Py of the particle

can only change by multiples of 2π~/L during its interaction with the lattice, i.e.,

2πn ∆p = ~, (2.51) y L

where n ∈ Z.

To verify this result, observe that, if the lattice is suﬃciently heavy and the initial wave

length of the charge is λ, the total momentum p = 2π~/λ of the particle should also be

unchanged. Then, while crossing the lattice, the charge is scattered by angles θn such that

∆py nλ sin θ = = , (2.52) n p L

20 which is the well-known expression for the discrete scattering angles in a diﬀraction experiment.

Now, suppose that each line on the lattice carries a magnetic ﬂux ΦB. Then, because of the 0 AB eﬀect, the angles in Eq. (2.52) are shifted by angles θn such that

0 ∆py qΦB λ sin θn = = n + . (2.53) p 2π~ L

This implies that the change in the y direction of the momentum is

0 qΦB ~ ∆py = p sin θn = 2πn + , (2.54) ~ L i.e., the modular momentum in the y direction is not conserved, diﬀerently from the scenario without a magnetic ﬂux on the lines. In fact, there is a change in modular momentum given by qΦ (p ) = B . (2.55) y mod L

It should be noticed that the value of the modular momentum in Eq. (2.55) refers to the kinematic momentum, i.e., the product of the mass of the particle by its velocity, in opposition to the canonical momentum, which is the canonical conjugated of the position. This distinction is important because, although both momenta coincide when the Hamiltonian is the one in Eq. (2.48), the two concepts diverge when the charge’s dynamics is governed, for instance, by the Hamiltonian in Eq. (2.29).

2.4 Instantaneous Change in the Transverse Velocity

The discussion at the end of the previous section suggests that there is a sudden change in the velocity of charges when the line connecting two wave packets crosses a ﬂux line carrying

21 a magnetic ﬂux ΦB. As shown in Ref. [57], this is, in fact, the case.

To see that, assume that a particle with charge q is prepared in the state

1 Ψ0(x, y) = √ ψ(x, y) + ψ(x + Lx, y + Ly) , (2.56) 2 where ψ is a wave packet and L~ = Lxxˆ + Lyyˆ is the separation between the centers of the wave packets. Suppose the packets evolve in time without changing the distance L = |L~ | between them until the particle crosses a ﬂux line placed at the origin of the coordinate system, meaning that the line that connects the centers of both wave packets crosses the ﬂux line. Also, let the Hamiltonian that governs the dynamics of the charge be

1 h i H = (P − qA (x, y))2 + P − qA (x, y)2 , (2.57) 2m x x y y where A~ = Axxˆ + Ayyˆ is the vector potential associated with the ﬂux line.

In this case, the modular velocity ~vmod of the particle can be deﬁned, and it is such that

2π m~v ≡ P~ − qA~ mod ~. (2.58) mod L

Furthermore, the modular velocity umod in the direction that connects the packets satisﬁes

L~ 2π mu ≡ P~ − qA~ · mod ~. (2.59) mod L L

Also, preparing for the results that will be presented next, observe that, for every holomorphic

22 function f and unitary U,

∞ † X † n f(U P~ · LU~ ) = U P~ · LU~ fn n=0 ∞ X † n (2.60) = U P~ · L~ Ufn n=0 = U†f(P~ · L~ )U.

~ ~ −iq R A~·d~`/ Then, letting f(P~ · L~ ) = eiP ·L/~ and U = e γ(~r) ~, and noting that

R R iq A~·d~`/~ −iq A~·d~`/~ e γ(~r) P~ · L~ e γ(~r) = P~ − qA~(~r) · L/~ ~, (2.61)

it can be concluded that

i P~ −qA~(~r) ·L/~ iq R A~·d~`−R A~·d~` / ~ ~ iq R A~·d~`/ ~ ~ e ~ = e γ(~r) γ(~r+L~ ) ~eiP ·L/~ = e τ(~r) ~eiP ·L/~, (2.62) where γ is an arbitrary path and τ is a segment of line with length L that goes from the center of packet 2 to the center of packet 1.

Now, suppose the ﬂux line carries zero magnetic ﬂux and denote the state of the charge at an instant of time t by 1 Ψ0(~r, t) = √ (Ψ1(~r, t) + Ψ2(~r, t)) . (2.63) 2

In this case, there is no change in modular velocity. However, if the ﬂux line carries a magnetic ﬂux ΦB, its state Ψ as a function of position and time can be written as

R ~ ~ R ~ ~ 1 iq γ A·d`/~ iq γ A·d`/~ Ψ(~r, t) = √ e 1 Ψ1(~r, t) + e 2 Ψ2(~r, t) , (2.64) 2 where γ1 refers to the trajectory of the center of packet 1 and γ2 refers to the trajectory of

23 the center of packet 2. With help of Eq. (2.62), this implies that

ZZ iq R A~·d~`/ ~ ~ heimumodL/~i = Ψ(~r, t)e τ(~r) ~eiP ·L/~Ψ(~r, t) dx dy

H ZZ 1 iq A~·d~`/~ = e γ Ψ1(~r, t)Ψ2(~r + L,~ t) dx dy 2 (2.65) 1 iq H A~·d~`/ = e γ ~ 2 1 = eiqΦenc/~, 2 where Φenc is the total magnetic ﬂux enclosed by the closed path. In Eq. (2.65), the Stokes’ theorem and the fact that

Z Z Z I A~ · d~` − A~ · d~` − A~ · d~` = A~ · d~` (2.66) γ1 γ2 τ(~r) γ were used. To see that Eq. (2.66) holds, it should be observe that, to be prepared in the superposition of packets given by Eq. (2.56), the charge must have started localized in a certain region. Then, γ1 and γ2 start at the same point.

Finally, because Eq. (2.65) implies that the modular velocity only changes with a variation in the magnetic ﬂux inside the area “covered” by the packets, there is an instantaneous change in the modular velocity umod when the line that connects the center of the two packets crosses the ﬂux line. Before and after that, there should be no change to the modular velocity.

It should be noticed that this result depends on the fact that the wave packets satisﬁes the property of being the translation of each other in a Cartesian system of coordinates. In a general case, the analysis of how the modular velocity changes is more complicated.

To conclude, the fact that the AB eﬀect can be seen as a change in the distribution of the velocity of a charge suggests that there may be a force acting on the particle. In fact, forces associated with the AB eﬀect were previously discussed in Refs. [56, 101–103]. This aspect will be further investigated in Chapter3.

24 2.5 Nodal Lines

A quantum particle confined to a finite region of space behaves drastically different from a classical particle. In particular, its energy becomes quantized. In fact, consider the problem known as the infinite well, which was likely [104] introduced by Mott [105] in 1930, and is typically one of the first examples presented in an introductory course on quantum mechanics. It consists of a particle confined in a region |x| < `/2, ` > 0, by the potential

  0 if |x| < `/2 V (x) = . (2.67)  ∞ if |x| ≥ `/2

In this case, the solutions of the Schr¨odingerequation vanish outside the region with zero potential. Moreover, inside the well, the eigensolutions of the Schr¨odingerequation are

r2 x 1 ϑ (x) = sin nπ + , (2.68) n ` ` 2 for every n ∈ N. Moreover, their associated energy is

2 2 2 n π ~ En = . (2.69) 2m`2

The quantization of energy is a general result of the conﬁnement of quantum particles that, then, holds not only for one-dimensional cases.

Here, following Refs. [47, 55], it is studied how the presence of a flux line affects the eigenstates of two-dimensional cavities. For that, consider a cavity with an impenetrable flux line on its interior. First, suppose there is no magnetic field traveling through the flux line. Then, the Hamiltonian of a free particle inside the cavity is given by Eq. (2.30), with the consideration that, besides being zero outside the cavity, the state that describes the particle

25 should also vanish at the ﬂux line. Now, any eigenfunction ψ0 of such a Hamitonian, i.e.,

H0ψ0 = Eψ0, (2.70) can be assumed to be a real function without loss of generality. Then, if the flux line carries ~ a magnetic flux ΦB, which is associated with a vector potential A, the Hamiltonian of the cavity becomes the one given by Eq. (2.29). Moreover, if the wave function ψ of a charge is an eigenfunction of such a Hamiltonian, it satisfies

Hψ = Eψ. (2.71)

With a similar approached used for Eqs. (2.31) and (2.32), it can be shown that there exists an eigenfunction ψ0 of H0 such that

R iq A~·d~`/~ ψ = ψ0e γ , (2.72) where γ denotes a path. In this case, Eq. (2.72) can be seen as generating a complex continuation rule of ψ0. In general, these extensions determine a Riemann surface [106, 107]. The rule in this case is

iWγϕ ψ(~rγ) = ψ0(~r)e AB , (2.73) where the winding number of γ, Wγ, is such that [108]

I ~ ~ A · d` = WγΦB. (2.74) γ

Eq. (2.73) should be read as a relation between the end points of a closed path γ in real space that starts in ~r and ﬁnishes in ~rγ in the Riemann surface.

In order to obtain a single-valued function, a sheet of such a manifold may be chosen. This

26 choice, however, introduces a discontinuity in the wave function in an arbitrary location. Nonetheless, it should be noticed that this discontinuity is only in the system’s phase. This approach of making ψ single-valued can be intuitively seen as an imposition of a periodic boundary condition on ψ0. With this view, instead of solving Eq. (2.29) for ψ, one solves

Eq. (2.30) for ψ0 with the extra condition that, in polar coordinates (ρ, θ),

−iϕ ψ0(ρ, θ + 2π) = e AB ψ0(ρ, θ). (2.75)

This is physically relevant because it implies that the absolute value of the wave function and, also, its probability density are independent of the choice of sheet.

Now, assume the ﬂux line inside the cavity carries a half-integer ﬂuxon, also called a semi-

ﬂuxon, i.e., ΦB = Φ0/2. This particular value is interesting because it makes the extension iϕ rule given by Eq. (2.73) real since ϕAB = π and, then, e AB = −1. As a result, ψ can be taken to be real.

Furthermore, because ψ changes its sign in a path which encircles a semifluxon, it must be zero somewhere in the path, regardless of the shape of the cavity. Therefore, by the continuity of the surface determined by ψ (or the continuity of its probability distribution in case a sheet is chosen), there must exist a nodal curve, i.e., a curve where ψ vanishes, starting at the flux line and finishing at the boundary of the cavity — or the position of another flux line carrying a semifluxon [47, 55].

It should be noted that linear combinations of eigenstates also possess nodal curves. More- over, nodal curves of eigenstates become straight lines according to the symmetry of the cavity.

To summarize and illustrate the main points of the discussion presented here, consider a circular cavity of unit radius with a semiﬂuxon at its center. In this case, the ground state

27 Figure 2.3: Representation of the real ground state of a circular cavity with a flux line carrying a semifluxon on its center. The three-dimensional plot on the left is the real Riemann surface associated with the “multivalued” wave function of Eq. (2.73) for ϕ = π. The color map defined on it associates colors to the magnitudes of ψ. This allows the drawing of the heat map of any arbitrarily chosen sheet of that surface, as represented on the right.

of the cavity when A~ = 0 is

ψ0(~r) = J1(ar), (2.76) where r is the radial coordinate, J1 is the Bessel function of the ﬁrst kind and parameter

1, and a ≈ 3.83 is the smallest positive zero of J1. Then, when the ﬂux line is carrying a semiﬂuxon, the real ground state can be expressed as

ψ(~r) = ψ0(r) cos(θ/2), (2.77)

where θ is the angular coordinate. Note that ψ is a “multivalued function”. On the left-hand side of Fig. 2.3, the Riemann surface associated with ψ is represented, while the heat map of any arbitrary choice of sheet is represented on the right-hand side. Observe the nodal line induced by the ﬂux line.

To conclude, it is worth mentioning that nodal curves appear in a variety of topological

28 systems, not only in systems with ﬂux lines. Recently, special attention has been dedicated to them, and they have been realized in multiple physical conﬁgurations [109–116].

2.6 Equivalence Principle

The AB eﬀect has an impact that goes beyond the study of electromagnetic systems. As an example of it, it will be shown, following Ref. [117], that it requires an extension of the classical equivalence principle in quantum physics.

The equivalence principle states that inertial and gravitational mass are equivalent. However, another way of seeing it is by stating that pseudo-forces that arise in non-inertial frames can be canceled by other forces and, then, physical systems in the non-inertial frames behave as if they were in an inertial frame.

Now, consider a laboratory given by a narrow ring with an inner radius R1 and an outer

radius R2, as represented by Fig. 2.4. Also, assume the lab is rotating with angular velocity ω and, to simplify the discussion, let all particles inside the lab have the same ratio of charge

over mass. Moreover, it should be noticed that the disc with radius R1 is massive.

Because of its rotation, the lab experiences two pseudo-forces, namely the centrifugal and the Coriolis force. Then, a quantum physicist, Alice, and a classical physicist, Bob, can detect that the lab is a non-inertial frame. However, suppose that another physicist, Charlie, decides ~ to trick Alice and Bob. For that, Charlie observes that the Coriolis force FC acting on an ~ ~ object with mass m and velocity (measured in the lab) ~v can be written as FC = m~v × C, ~ ~ ~ where C is the ﬁeld associated with FC. Moreover, C satisﬁes

∇~ · C~ = 0, (2.78)

~ ~ which implies that FC is given by a ﬁeld that is the rotation of a vector potential VC. Also,

29 Figure 2.4: Representation of a laboratory given by a narrow ring. (a) A classical physicist, Bob, and a quantum physicist, Alice, can verify whether the laboratory (pink region) is spinning or not. (b) However, when an electromagnetic ﬁeld (blue region) is added to cancel the non-inertial forces associated with the spinning inside the laboratory, Bob can no longer know he is not in an inertial frame. (c) Nonetheless, Alice can detect the spinning with a particle in a superposition (yellow clouds) encircling the disk internal to the laboratory (gray area), where the forces were not canceled by the electromagnetic ﬁeld, which causes a shift in the interference pattern proportional to the angular velocity of the disk. if F~c is the centrifugal force,

∇~ × F~c = 0, (2.79) i.e., F~c can be written as the gradient of a scalar potential. Therefore, there exists a parallel between the centrifugal and the Coriolis forces with electromagnetic forces.

Charlie, then, decides to apply an electromagnetic ﬁeld inside the lab in order to remove the pseudo-forces. Because of this, Bob is not able to detect that the lab is a non-inertial frame. In fact, any experiment he performs with classical particles will give results that correspond to measurements made in inertial systems.

Nevertheless, because of the AB eﬀect, Alice can conduct an interference experiment with a quantum particle and detect that the lab is rotating. The reason is that, although Charlie canceled the pseudo-forces inside the lab, these pseudo-forces were not canceled in the interior of the disk with radius R1. Then, if a quantum particle encircles the disk in a superposition of packets traveling in the clockwise and counterclockwise directions, the interference pattern

30 detected by Alice will be shifted by a phase proportional to

I Z ~ ~ ~ ~ 2 m VC · d` = m C · dS = 2πmR1ω, (2.80) D1 where D1 is the disk with radius R1.

The scenario considered here shows that the equivalence principle needs to be modiﬁed in quantum physics, and this is an example of how the implications of the AB eﬀect can reach areas that go beyond the dynamics of quantum charges.

31 3 Magnetic Forces from the Aharonov-Bohm Eﬀect

As discussed in the previous chapter, a force can be associated with the Aharonov-Bohm (AB) eﬀect, even though it does not result from a local interaction between the charge and the magnetic ﬁeld. In this chapter, the main results from Refs. [66, 67] are presented, showing that such a force corresponds, in a certain sense, to a magnetic force.

Specifically, it is discussed how an infinite lattice of flux lines acts as an energy barrier for quantum charges and that two flux lines can be used to confine low-energy charges in a sector of a long cavity. More generally, it is shown that grids of flux lines can emulate continuous two-dimensional magnetic fields. This allows the obtaining of the Landau levels in a uniform grid of flux lines. This result also suggests a semi-classical model where the force associated with a grid of flux lines can be seen as quantum counterparts of forces from classical continuous two-dimensional magnetic fields.

3.1 Continuous Magnetic Fields in Classical Physics

In this section, a simple example of a magnetic ﬁeld as an energy barrier in classical physics is presented. This example will be further explored in the context of quantum mechanics in the next two sections.

Consider a (penetrable) wall parallel to the y axis with a uniform magnetic ﬁeld B~ = Bzˆ and a point particle with mass m and charge q traveling in the x direction, as represented

in Fig. 3.1. Moreover, let ΦB = BLw be the magnetic ﬂux associated with any arbitrary region with length L and width w. When the particle is inside the wall, a force F~ = ~v × B~ , where ~v is the particle’s velocity, acts on it. Then, using Newton’s second law, it can be

32 Figure 3.1: Classical charges (yellow dots) traveling towards a uniform magnetic ﬁeld wall (blue region) given by B~ = Bzˆ and such that any rectangular region with length L and width w encloses a ﬂux ΦB = BLw. Their trajectories are represented in red. In (a) and (b), low-energy particles cannot cross the wall, while a high-energy particle is scattered by it in (c). In general, if a particle passes through the wall, its transverse kinematic momentum is changed by −qΦB/L.

checked that the charge is deﬂected by the ﬁeld into a circular arc of radius

mvw R = , (3.1) |qφB|

where Φ φ = B (3.2) B L

is the magnetic ﬂux per unit of transverse length. Then, if the speed v = k~vk of the particle is such that R < w, i.e., if the magnitude of the particle’s kinematic momentum px = mv is such that

px < |qφB| , (3.3) the particle is reﬂected by the wall.

Note that, here, and everywhere in the present chapter, there is no bound on the z component of the charge’s kinematic momentum (or on the energy associated with it), i.e., it can be arbitrarily large and remain conserved without aﬀecting the results.

33 Now, if the width w of the wall is reduced, while the magnitude of the field B is simulta- neously increased in a way that φB remains constant, the maximum speed v for which a given charge is reflected by the wall remains unchanged. This continues to hold even in the limit w → 0, i.e., when the magnetic field wall becomes widthless. Therefore, the barrier is characterized entirely by the magnetic flux per unit of length φB associated with the wall.

Furthermore, Eq. (3.1) is valid even if the particle is incident at an arbitrary angle in the xy plane, as represented in Fig. 3.1(c). Then, it can be checked with a simple geometric analysis that, in general, the wall reﬂects all those charges if 2R < w, i.e., if their planar

q 2 2 kinematic momentum mv = px + py is such that

q |qφ | p2 + p2 < B . (3.4) x y 2

Finally, if the charge crosses the wall in a time interval ∆t, the change in transverse kinematic momentum is Z ∆py = −qB vx(t)dt = −qBw = −qφB, (3.5) ∆t i.e., the change in the transverse kinematic momentum is independent of the angle of incidence and the velocity of the particle. It can also be verified that, when the charge is reflected, there is no change in the magnitude of its transverse kinematic momentum after the interaction with the wall: the angle of incidence equals the angle of reflection.

3.2 Continuous magnetic ﬁelds in quantum mechanics

In this section, the direct quantum analog of the result from the previous section will be investigated. For simplicity, only the inﬁnite widthless magnetic wall will be considered.

First, the reference frame is such that the wall is placed on the y-axis, and the particle starts

34 on the left-hand side. A possible choice of vector potential for this scenario is

~ A = φBΘ(x)ˆy, (3.6) where Θ is the Heaviside step function, i.e.,

  0 if x < 0 Θ(x) = . (3.7)  1 if x ≥ 0

In this gauge, the Hamiltonian of the system can be written as

1 h i H = P 2 + P − qφ Θ(X)2 , (3.8) 2m x y B which implies that the initial average energy of the charge can be computed as

1 hEi = hP 2i + hP 2i (3.9) i 2m x i y i

and its ﬁnal average energy, in case it crosses the lattice, is

1 2 2 2 2 hEif = hPx if + hPy if + q φB − 2qφBhPyif 2m (3.10) 1 = hP 2i + hP 2i + q2φ2 − 2qφ hP i , 2m x f y i B B y i

where it was used the fact that the canonical transverse momentum is conserved, i.e., hPyii = 2 2 hPyif and hPy ii = hPy if , which follows from H,Py = 0.

To avoid possible confusion due to the asymmetry in A~, note that, if the particle had started

on the right-hand side, its initial kinematic momentum would be given by mvy = Py − qφB. Then, if the particle started with zero transverse kinematic momentum, for instance, its

initial canonical transverse momentum would be such that hPyii = qφB. Again, the canonical

35 momentum would be conserved, which implies that the particle’s final transverse kinematic momentum after it crossed the wall would be, on average, qφB. This verifies the symmetry of rotation of the xy plane by 180◦, which justifies the restriction to particles starting on the left-hand side of the wall.

With the requirement of change in the average transverse kinematic momentum, it is possible to conclude that a charge cannot completely cross the wall whenever hEii < hEif,min. To be precise, higher-energy components of the wave packet associated with the charge pass through the wall, while lower-energy components are reflected. Generally, the probability of finding the particle goes to zero in the far-field on the side opposite to the one it started.

Then, if the particle is incident with average (canonical and kinematic) momentum hPxii in the x axis and hPyii in the y axis, its minimum average energy after crossing the wall is obtained when hPxif → 0, and it cannot completely pass it if

2 2 2 hPx ii < q φB − 2qφBhPyii. (3.11)

It follows that, if the particle is perpendicularly incident, i.e., hPyii = 0, the above condition becomes

hPxii < |qφB| , (3.12)

2 2 where it was used the fact that hPx i ≥ hPxi . Observe that Eq. (3.12) is analogous to

Eq. (3.3). Moreover, if hPyii 6= 0, Eq. (3.11) only admits solution if its right-hand side is positive, i.e., if hPyii < |qφB| /2, then, the condition for a charge to be at least partially reﬂected by the wall, regardless of the angle of incidence, is

q |qφ | hP i2 + hP i2 < B , (3.13) x i y i 2 which is analogous to Eq. (3.4).

36 3.3 Emulating Continuous Magnetic Fields with Distributions of Flux Lines

In this section, the widthless magnetic wall is replaced by flux lines — or infinitely thin solenoids. In two dimensions, these lines are point objects. Without loss of generality, the magnetic flux on each line is assumed to be positive. This can always be achieved with a rotation of the referential system. Also, recall that the influence of each flux line in the dynamics of quantum charges is invariant under the addition of a fluxon, i.e., the magnetic

flux Φ0 defined in Eq. (2.38). Because of this, the magnetic fluxes can be taken to be limited

to the interval [0, Φ0).

The continuous magnetic wall is replaced with a lattice of ﬂux lines with spacing L, with each line carrying a magnetic ﬂux ΦB, as represented in Fig. 2.2. As discussed in the previous chapter, an incident charge represented by a plane wave has the modular part of its transverse kinematic momentum changed according to Eq. (2.55). What is most important for the present analysis is the fact that this implies that there is a minimum change in the transverse kinematic momentum of the charge, which is given by ∆py min = |qφB| if

ΦB ≤ Φ0/2, and 2π~/L − qφB if ΦB > Φ0/2. Observe that the latter corresponds to the minimum deﬂection if −Φ0/2 < ΦB < 0. Because of this extra symmetry, hereby, only

magnetic ﬂuxes in the interval (0, Φ0/2] are considered.

(i) (i) (i) Let the charge start with average kinematic momentum pi = px xˆ + py yˆ. Then, if py = 0,

the particle acquires a transverse kinematic momentum of at least −qφB. Therefore, if (i) (i) px < |qφB| (Eq. (3.3)) is satisﬁed, the charge is reﬂected. Moreover, if py 6= 0, its transverse kinematic momentum can, in principle, decrease in magnitude after it crosses the

(i) lattice. However, if py < ∆py min /2, the magnitude of the ﬁnal transverse kinematic

(i) momentum of the particle cannot be smaller than py − ∆py min , which is still greater

(i) than py . In other words, if Eq. (3.4) is satisﬁed, the charge must bounce oﬀ the lattice of

37 ﬂux lines. In conclusion, the lattice constitutes an energy barrier similar to the continuous magnetic wall. Interestingly, Eq. (3.4) does not depend explicitly on ~. However, because

ΦB is upper-bounded by Φ0 = 2π~/q, for any ﬁxed lattice spacing L, φB = ΦB/L → 0 when

~ → 0. However, L can be adjusted so that L → 0 and φB is constant in that limit. This shows that, in specially designed conﬁgurations, consequences of the AB eﬀect can still hold in the classical limit.

3.4 Topological Bound States

In this section, it is shown that flux lines can be used to create bound states. However, to continue the parallel between discrete and continuous distribution of fields, the cases with continuous magnetic fields are considered first.

In classical physics, it was studied in Section 3.1 that widthless magnetic walls behave like mirrors for low-energy charges. If the wall is placed inside a long rectangular cavity, as represented in Fig. 3.2(a), it continues to behave like a mirror. Furthermore, if a low-energy charge is placed between two of those walls, independently of how close they are from each other, it remains trapped in that region, as shown in Fig. 3.2(b).

A similar result holds in a quantum treatment of the problem. An argument close to the one presented in Section 3.2 can be given. For that, consider a quantum charge traveling towards a magnetic wall, as represented in Fig. 3.2(c). Choosing a gauge for which the vector potential inside the cavity is written as Eq. (3.6), it is possible to conclude that there should be an increase in the particle’s average kinematic energy associated with the y direction after it completely crosses the wall. Then, if its initial average energy is suﬃciently small, the charge is at least partially reﬂected.

Now, a quantum charge can also be trapped by two walls of magnetic ﬁeld, as illustrated

38 Figure 3.2: Representation of particles inside a cavity. A Classical particle (yellow dot) can be (a) reflected trapped and (b) trapped by magnetic walls (blue lines). The red curves represent the particle’s trajectory. For a quantum charge q (yellow cloud) inside a cavity, the magnetic walls also (c) work as an energy barrier, and (d) can trap the particle. This effect persists in the quantum case even if the wall is replaced by a flux line (blue dot), as shown in (g) and (h). In particular, if the flux line carries a semifluxon, nodal lines are induced in the wave functions of charges, as illustrated by the green lines in (e) and (f). in Fig. 3.2(d). As discussed in Section 2.5, the energy of a quantum particle confined to a cavity becomes quantized. There, the case of an infinite well was given as an example. The rectangular cavity of interest here is a two-dimensional version of that problem. In fact, because the Schrödinger equation becomes separable, the solution to this problem is similar to the one-dimensional case. The overall eigenstates are products of the eigenstates of each dimension and their energies are the sum of the energy associated with each direction.

While these ideas did not play a role in the discussion of the scenario represented in Fig. 3.2(c), they are fundamental in the case where a particle is conﬁned by two magnetic walls inside a long rectangular cavity. In fact, letting L be the width of the cavity and D be the distance between the walls, the minimum average energy of a particle that starts in the

39 region between the walls must have at least an average energy of

π2 2 π2 2 hEi = ~ + ~ , (3.14) i,min 2mL2 2mD2 i.e., the minimum energy of a particle inside a two-dimensional box with lengths D and L. Moreover, the amount of average energy necessary for a charge to completely cross one of the walls is greater than or equal to

π2 2 q2φ2 hEi = ~ + B . (3.15) f,min 2mL2 2m

Hence, if |qφB| > π~/D, i.e., if the separation D between the walls is such that

π D > ~ , (3.16) |qφB| it is possible to have a low energy particle that stays (at least partially) trapped in the region between the two walls. Observe that, in general, the particle will have a non-trivial dynamics inside the region between the walls. In any case, the result just presented implies that the walls create bound states inside the cavity. Moreover, diﬀerent from the classical case, there is a minimum distance between the walls required for the possibility of having a particle conﬁned by them. However, in the classical limit, i.e., when ~ → 0, the value of this minimum distance goes to zero.

Finally, the case where the magnetic walls are replaced by flux lines, which is the main focus of this section, is now considered. Once again, consider a rectangular cavity with width L equipped with two flux lines placed with a distance D from each other on the long symmetry axis of the cavity, as shown in Fig. 3.2(e). Then, the following question can be asked: do the flux lines work similarly to the walls of magnetic field just considered? Or, more specifically, do they create bound states inside the cavity?

40 To approach this problem, first consider flux lines carrying a semifluxon each, i.e., a magnetic

flux given by ΦB = Φ0/2. This simplifies the analysis because, as studied in Section 2.5, semifluxons induce nodal lines in the wave function of charges. In this particular configuration, the symmetry of the problem implies that the nodal lines associated with the flux lines for eigenstates of the cavity should be as illustrated by the green lines in Figs. 3.2(e) or 3.2(f). To see that, note that the Hamiltonian of the cavity remains physically invariant if the cavity is rotated by 180◦ about its long axis of symmetry. The difference between the two configurations (the original and the rotated one) is the direction of the semifluxon.

Then, the flux lines differ by an entire fluxon Φ0, which does not affect any physical system.

A charge whose wave function is a packet moving towards the right-hand side, initially conﬁned in the sector delimited by the two ﬂux lines, and given by a product state ψ = ψxψy, with r 2 πy ψ = cos , (3.17) y L L

has a minimum average energy hEii,min given by Eq. (3.14).

In this setup, the eigenstates that contribute to the eigendecomposition of the particle must be the ones whose nodal lines are in the configuration shown in Fig. 3.2(f). Thus, a charge that completely escapes the region between the flux lines will present a nodal line at the center of the y direction. Then, the state with minimum average energy after a particle completely escapes the flux lines corresponds to the first excited state of the y direction, i.e.,

2π2 2 hEi = ~ . (3.18) f,min mL2

As a consequence, if the charge starts with average energy smaller than this value, which may be the case if π2 2 π2 2 2π2 2 1 ~ + ~ < ~ ⇒ D > √ L, (3.19) 2mL2 2mD2 mL2 3

41 it cannot completely leave the region. Even if parts of its original wave function associated with higher energies move past the flux lines, parts of it bounce back. This remaining part remains trapped in the region between the flux lines and, therefore, is given by a linear combination of eigenstates with nodal lines outside the region between the flux lines. Thus, there exist eigenstates with low energy associated with the configuration presented in Fig. 3.2(f) that are mostly confined in sector delimited by the flux lines.

This result is valid even if the cavity is infinitely long, i.e., if an open wave-guide is considered. Moreover, even though the confinement of these systems depends on the geometry of the cavity (it must be relatively narrow, for instance), they would not exist if it was not for the AB effect. Because of this, they are called topological bound states.

Also, the continuity of the probability density associated with the wave function implies that the topological bound states have tails outside the region between the ﬂux lines. However, position measurements of topologically bounded charges must reveal, with substantial probability, that they are in the region between the solenoids, and the charge should not be found in the far (outer) ﬁeld.

Now, what happens if each flux line carries a magnetic flux ΦB different from a semifluxon? To answer this question, consider the cavity of Fig. 3.2(g) with a single flux line carrying a

magnetic flux ΦB placed, say, at the origin of a frame of reference whose x axis is parallel to the long symmetry axis of the cavity. Assume the cavity has a long extension to the left and the right of the flux line. Like in the case with semifluxons, let the charge start in a separable state ψ = ψxψy, where ψy is given by Eq. (3.17) and ψx is a state with low average energy in the direction of motion.

If ΦB = 0, ψ returns to its initial shape after the packet has completely passed the ﬂux

line. Now, if ΦB = 1, then ψ is slightly disturbed as the charge passes it, since it must present a phase discontinuity after the charge completely encircles the ﬂux line. To treat the

42 problem, consider a gauge for which the vector potential A~ associated with the ﬂux line is given by ~ A(x, y) = ΦBΘ(x)δ(y)ˆy. (3.20)

Then, after crossing the flux line, ψ turns into ψ0, which must have a phase discontinuity at y = 0. Since the magnetic flux is considered to be sufficiently small, ψ0 ≈ ψ and, while 0 0 ψy ≈ ψy, ψy has a phase discontinuity of at y = 0. To see that this phase discontinuity implies that the flux line creates an energy barrier, observe that the change in the average energy associated with the y direction can be approximated as

Z γ 1 ∗ 2 ∆hEi ≈ lim ψy Py − qδ(y) ψy dy 2m γ→0 −γ Z γ (3.21) 1 2 2 2 2 = q lim δ(y) ψy dy. 2m γ→0 −γ

Because ψy was already prepared with the minimum possible energy in the y direction, this change of energy necessarily implies an increase in the average energy associated with that

direction. As a result, as long as the initial average energy of ψx is smaller than a certain threshold, the charge is at least partially reﬂected by the ﬂux line.

The problem, now, becomes the quantification of the minimum amount of extra energy 0 associated with ψy after the particle crosses the flux line. For that, consider, once more, the lattice of flux lines represented in Fig. 2.2 and let the initial state of the charge be

ψ = ψxψy, where ψy is given by Eq. (3.17), which has lines of zeros at y = nL + L/2, evenly spaced between the fluxes of the lattice, as represented by the gray lines in Fig. 2.2. For the incoming particle, the dynamics will be unchanged if infinite cavity walls (not magnetic) are added along the nodal lines, to the left of the lattice. Now, when the particle passes the flux lines, these walls end, and the dynamics goes back to the case of free diffraction, where Eq. (2.55) applies. Moreover, adding in the cavity walls on the right-hand side of the lattice to the initial Hamiltonian can only increase the minimum energy associated with the

43 y direction. In other words, the free case gives a lower bound on the energy for the stacked

cavities case. Hence, it is possible to conclude that the energy increase associated with ψy 2 2 after the charge crosses the ﬂux line corresponds to at least the amount q φB/2m. Thus, low-energy particles cannot cross the ﬂux line.

With this in mind, consider a cavity with two flux lines separated by a distance D. If a charge starts in the region between the fluxes, the minimum amount of average energy it can have is given by Eq. (3.14). After crossing the flux line, the charge’s minimum amount of average energy is expressed in Eq. (3.15). As with the case of two magnetic walls inside a cavity, it can be concluded that, if Eq. (3.16) is satisfied, there exist topological bound states in the sector of the cavity delimited by the flux lines.

Finally, observe that, if each wall has a magnetic flux greater than a semifluxon (i.e., Φ0/2), then it should be replaced by multiple flux lines, instead of a single one. In any case, the results discussed here should still hold.

3.5 Landau Levels

So far, in this chapter, the region with magnetic field was taken to be widthless. But this was done merely for convenience. In fact, as it will be shown in this section, the effects of two-dimensional magnetic fields can be emulated by two-dimensional grids of flux lines. In particular, approximations of the Landau levels will be obtained in the presence of a uniform two-dimensional grid of flux lines.

In classical physics, a classical charge inside a region with a uniform magnetic ﬁeld travels in a circular motion. In quantum mechanics, these circular orbits are also present in the dynamics of charges. However, they are quantized, and each level of energy is called a Landau level. To see that, consider a particle in a uniform magnetic ﬁeld B~ = Bzˆ. Then,

44 in the gauge known as the Landau gauge, for which the vector potential is A~ = Bx, the Hamiltonian of the particle is given by

1 h i2 H = P 2 + P − qBX2 . (3.22) 2m x y

Because H,Py = 0, the transverse momentum can be replaced by its eigenstate ~ky. With that, the Hamiltonian is simpliﬁed to

1 h i2 H = P 2 + k − qBX2 , (3.23) 2m x ~ y which can be identiﬁed as the Hamiltonian of the one-dimensional harmonic oscillator. Har- monic oscillators play a fundamental role in the study of systems near their equilibrium. Also, they correspond to a one-dimensional projection of two-dimensional circular motions. The fact that Eq. (3.23) is the Hamiltonian of a harmonic oscillator indicates that the particle travels on a circular orbit, although the asymmetry of the chosen gauge does not seem to corroborate with this conclusion at ﬁrst glance. However, a more careful analysis with, for instance, the choice of the symmetric gauge (A~ = (−yxˆ + xyˆ)B/2) can reveal the circular characteristic of the solution.

The idea, now, is to show that an approximation of the Landau levels can be obtained with the use of flux lines. To see that, consider a region with constant magnetic field B~ = Bzˆ. Also, let this region be divided into squares with length L, each with a flux 2 ΦB = BL < Φ0/2 — or a magnetic flux per unit of transverse length φB = B/L. Now, replace each square by a flux line with magnetic flux ΦB. Then, it is still possible to obtain the Landau levels with this two-dimensional square grid of flux lines with spacing L. In the

45 singular gauge, the Hamiltonian of a charge can be written as

  2 1 X H = P 2 + P − q A (X,Y )  , (3.24) 2m  x  y ns   n,s∈Z where Ans(X,Y ) = ΦBΘ(X −nL)δ(Y −sL) is the vector potential associated with each flux line. The solutions of this Hamiltonian cannot be easily found. However, the Hamiltonian can be simplified by using the fact that, for each vertical layer, the average effect of the flux lines is a change of qφB in vertical momentum. Hence, Eq. (3.24) can be approximated as

  2 1 X H = P 2 + P − qφ Θ(X − nL)  . (3.25) 2m  x  y B   n∈Z

Now, because the new expression for the Hamiltonian commutes with the canonical transverse momentum Py, it is possible to replace this operator by its eigenvalue ~ky. Then,

  2 1 X H = P 2 + k − qφ Θ(X − nL)  . (3.26) 2m  x ~ y B   n∈Z

One can easily see that the Hamiltonian in Eq. (3.26) is formally an approximation of the one-dimensional harmonic oscillator. In fact, if L 1, the term φ P Θ(X − nL) can B n∈Z be approximated as BX. This shows that, indeed, the Landau levels can be recovered with the use of a two-dimensional grid of ﬂux lines.

3.6 Semi-Classical Toy Model

Recall that the magnetic ﬂux ΦB associated with a ﬂux line vanishes in the classical limit.

However, this does not necessarily imply that φB also vanishes. Indeed, as it will be discussed in this section, it is possible to take the distance L between the ﬂux lines to zero in the

46 Figure 3.3: Schematic representation of a semi-classical theory where a region with an arbitrary continuous distribution of magnetic field in the z direction that does not depend on the z coordinate (blue region) is replaced by a discrete distribution of flux lines (blue dots). The region can be split into infinitesimal areas, each with constant magnetic fields, as represented in the rectangular zoomed-in cut-away section. These infinitesimal areas can be replaced by a grid of flux lines, as shown in the elliptical zoomed-in cut-away section. The charge (yellow object) is assumed to have a spread much smaller than the infinitesimal areas. classical limit in a way that keeps φB constant. In this case, the minimum deflection does not vanish. It seems, then, that the AB effect generates a classical force.

In fact, under certain seemingly reasonable assumptions, the classical magnetic force in an arbitrary continuous field B~ (x, y) = Bz(x, y)ˆz can be seen as arising from the topological AB force. To show this, the magnetic field has to be first broken up into differential rectangles of area dx · dy, each with flux dΦB = Bz(x, y)dxdy, as illustrated in Fig. 3.3. Then, the uniform magnetic field Bz in the differential region dx · dy is replaced by an M × N grid of lines with fluxes dΦB/MN.

Now, consider a quantum charge q spread over a region much smaller than dx · dy (but large enough to be diffracted by some of the flux lines) and incident on one of the infinitesimal cells with average velocity

~v = vxxˆ + vyyˆ + vzz.ˆ (3.27)

Also, assume that the AB force acts on the wave function of the particle when it is crossing

47 the lattice, which is consistent with the discussion in Section 2.4. Then, neglecting the terms

with ~ in Eq. (2.54), which vanish in the classical limit, the change in kinematic momentum

per vertical layer of N ﬂux lines in the y direction amounts to −qBz(x, y)dx/M. Similarly, the change in kinematic momentum in the x direction per horizontal layer of M ﬂux lines is qBz(x, y)dy/N. Then, if the center of the charge spread crossed n1 ≤ N vertical layers and

n2 ≤ M horizontal layers while passing over the dx · dy inﬁnitesimal cell, the total change in kinematic momentum can be approximated as

d~p = dpxxˆ + dpyyˆ (3.28) n n ≈ qB (x, y) 1 dy xˆ − qB (x, y) 2 dx y.ˆ z M z N

Noticing that the particle’s average velocity is kept approximately constant in each inﬁnites-

imal cell, i.e., vx ≈ (n2/N)(dx/dt) and vy ≈ (n1/M)(dy/dt), where dt is the amount of time the center of the charge’s distribution remains in the cell, a simple application of the chain rule gives dp dpy F~ = x xˆ + yˆ = qB v xˆ − qB v y,ˆ (3.29) dt dt z y z x

which correspond to F~ = q~v × B.~ (3.30)

Finally, taking the classical limit where the particle spread reduces to zero, ~v becomes the

classical velocity, and ~ goes to zero, the classical force F~ experienced by a point charge q in a magnetic ﬁeld B~ arises purely from the topological AB force.

3.7 Discussion

The results discussed in this chapter show a certain equivalence between the quantum and the classical treatment of the dynamics of charges in the presence of continuous magnetic

48 fields. This, on itself, is a manifestation of how special magnetic fields are. To evidence it, it is interesting to compare the magnetic wall with a usual potential barrier. First, recall that, even though the energy barrier associated with the magnetic walls is, in general, dependent on the angle of incidence, there exists a value (|qφB|/2) for which any low-energy charge is reflected by the wall. For simplicity, take that value as the value of the energy barrier imposed by the wall. In that case, if the wall was replaced by a scalar potential given by

qφ V (x) = B δ(x), (3.31) 2 or even by   0 if x < 0 or x ≥ w V (x) = , (3.32) |qφB|  2 if 0 ≤ x < w the potential wall would behave like a mirror for low-energy charges for particles only in the classical treatment of the problem. For quantum systems, those scalar barriers are not enough to reﬂect low-energy particles. This is because of a phenomenon known as quantum tunneling, discovered in 1927 by Hund [118].

The idea of this phenomenon is that, even though the particle does not have enough energy to overcome the barrier, slowly it starts “leaking” on the other side. Therefore, the particle can be found in the far-ﬁeld. The potential barriers only reﬂect low-energy particles in the limit where w → ∞, i.e., when the potential is uniformly applied everywhere on the opposite side of incidence of the particle.

Besides this perspective on magnetic fields, the results presented in this chapter also show that the AB effect enables the construction of energy barriers with lattices of flux lines. For magnetic fluxes between zero and Φ0/2 = π~/q, these lattices and thin continuous magnetic walls behave alike. Their similarity, however, vanishes outside that interval because of the periodicity discussed in Section 3.2. Nonetheless, this is not a significant limitation of our

49 results. In fact, if a magnetic wall has a flux ΦB > Φ0/2 associated to any L, there is a 0 0 length L < L such that the magnetic flux ΦB associated with a region of that length is 0 ΦB < Φ0. Thus, in general, any widthless uniform magnetic wall with φB can be replaced 0 0 0 by a lattice of flux lines with ΦB = φBL ≤ Φ0/2 = π~/q, where L is the spacing of the lattice.

Additionally, it was discussed how these results could be extended to the case where the magnetic walls have some width, and the lattices are two-dimensional grids of ﬂux lines. In particular, in Section 3.5, it was shown how approximations of Landau levels could be obtained with distributions of ﬂux lines.

It is important to note that, before this work, Andrei Shelankov studied wave functions with ﬁnite width in the presence of ﬂux lines using paraxial analysis and obtained some of the results presented here [119, 120]. However, this work not only uses a more straightforward approach to generalize Shelankov’s results, but also introduces new ideas, like the topological bound states, for instance.

Finally, the basic scheme for a semi-classical theory presented in Section 3.6 suggests that the AB eﬀect in quantum mechanics may be the fundamental source of the classical magnetic force. Further developments of this semi-classical model where the spread of the particle is introduced as a free parameter are necessary to shed more light on this quantum-to-classical transition.

50 4 Complex Vector Potentials in Pre and Post-Selected Systems

Up until now, the AB effect has been considered in scenarios where the source of the electromagnetic field is not quantized. This chapter, however, presents the results from Ref. [68], which studies quantum charges encircling a quantized magnetic flux. It is shown that the state of the particle after it surrounds a flux with small uncertainty is approximately the final state of a particle enclosing a region with the average magnetic flux. Furthermore, it is proved that, if a post-selection of the magnetic flux is considered, the magnetic vector potential is, in general, complex-valued.

4.1 Weak Measurements and Weak Values

As already mentioned, if a property O of a system in a state |ψi is measured, the system will be found in an eigenstate |oi of O with probability given by

p(o) = |ho|ψi|2. (4.1)

However, for a measurement to happen, a measurement device has to interact with the system of interest, and the result is, then, read from the device (which is also called the pointer). Thus, the measurement of the system is made indirectly.

The model of how the interaction between the device and the system takes place was introduced by von Neumann [24]. He considered the following interaction Hamiltonian

Hint(t) = h(t)O ⊗ P, (4.2)

51 where O is the observable of interest (that acts in the Hilbert space (HS) Hs of the system),

P is the momentum operator that acts in the HS Hd of the device, and h is a function of

time that is non-zero from t0 to t1, the interval of time where the measurement takes place.

The interest here is on interactions that happen fast enough when compared to the evolution

of the system and the apparatus. In this case, Hint is the eﬀective Hamiltonian during the measurement. These are called impulsive measurements. Hence, the unitary evolution is

R t 0 0 −i t Hint(t )dt /~ U(t; t0) = e 0 . (4.3)

To see how the device ends up with information about the system, let the initial state of the device be Z |ξi = ξ(x)|xidx. (4.4)

Moreover, let the initial state of the system be

Z |ψi = ψ(o)|oido, (4.5)

where |oi are eigenstates of the observable O. Then, writing Ψ(t0)i = |ξi ⊗ |ψi,

where g ≡ R t1 h(t) dt. t0

52 More generally, in the Heisenberg picture,

Z t † 0 0 I ⊗ X(t) = U (t)(I ⊗ X(t0) U(t; t0) = I ⊗ X(t0) − h(t ) dt O(t0) ⊗ I. (4.8) t0

Hence,

hX(t1)i = hX(t0)i − ghO(t0)i, (4.9)

where it was used that ∂F [X, f(P )] = i (P ), (4.10) ~∂P

which follows from Eq. (2.8). Eq. (4.9) shows that, in the average over many trials of the experiment, the measurement device is shifted by an amount that is proportional to the average of the observable of the measured system. While Eq. (4.9) also holds for density matrices, for simplicity, only pure initial states are considered hereby.

It can be observed that the constant g is associated with how well separated the eigenvalues of the observable will be in the reading of the device, i.e., how well the device can resolve the observable O. If g is large enough, the measurement is sharp, and the experimentalist has high conﬁdence about the state of the system of interest after the measurement is complete. However, if g is small enough, the measurement will not provide enough information for the experimentalist to know with high certainty the state of the system of interest.

It turns out that, while a large g can highly disturb the system, the case of a suﬃciently small g keeps the system relatively intact. If fact, from Eq. (4.7), it holds that the state of the system after the interaction with the device is

53 Now, assuming ξ is analytic,

dξ ξ(x − g(o − o0)) ≈ ξ(x) − g(o − o0) (x) (4.12) dx as long as |g(o − o0)| 1 for every o and o0 such that ψ(o) 6= 0 and ψ(o0) 6= 0. In this case,

ZZZ 0 0 0 ρs ≈ ψ(o)ψ(o )ξ(x)ξ(x)|oiho | do do dx ZZ = ψ(o)ψ(o0)|oiho0| do do0 (4.13)

= |ψihψ|.

This means that the ﬁnal state is almost a product state if g is much smaller than the inverse of the spreading of the measured system. In particular, the measured system remains approximately unchanged. This is called a weak measurement. With that, the case where the system is highly disrupted is, then, called a strong measurement.

Furthermore, in a weak measurement,

0 −ig(o−hOi)⊗P/~ |g(o − o )| ~ ⇒ |g(o − hOi)| ~ ⇒ e ≈ Id (4.14) for every o and o0 such that ψ(o) 6= 0 and ψ(o0) 6= 0, and the ﬁnal state of the joint system is such that

54 Figure 4.1: Eﬀect of measurements in the measurement device. The initial distribution of the measurement device (pink area) is a Gaussian with zero mean and a standard deviation of σ = 0.4. The purple curve represents its distribution after the measurement is complete. It should be noticed that weak measurements (small g) almost does not disturb the device, while strong measurements (large g) drastically changes its distribution.

As an example, consider the case where the initial state of the device is a Gaussian centered at the origin, i.e., 1 1 x2 ξ(x) = √ exp − . (4.16) σ 2π 2 σ

Also, suppose the system of interest is a qubit, i.e., a quantum system associated with a complex HS Hs with dimension 2, in the state

1 r2 |ψi = √ |0i + |1i, (4.17) 3 3 where |0i and |1i are an orthonormal basis of Hs. Moreover, consider the measurement of an observable σz such that σz|0i = |0i and σz|1i = −1. Thus, Eq. (4.7) becomes

55 Since the ﬁnal state of the device ρd can be computed as ρd = Trs|Ψ(t1)ihΨ(t1)|, the ﬁnal probability distribution of it is

2 2 1 − x−g 1 − x+g |ξ|2(x) = hx|ρ |xi = e σ + e σ . (4.19) d 6σ2π 3σ2π

This state is compared with the initial state of the device (the square of Eq. (4.16)) in Fig. 4.1 for various choices of g and σ = 0.4. It should be noticed that, for weak measurements, as a result of the measured system remaining approximately undisturbed, the probability distribution of the pointer is also kept almost unchanged. This implies that little information can be extracted from a single measurement since the device does not have enough resolution to give a precision of how much its average was shifted. However, observe that repetitions of the measurement eﬀectively decrease the variance associated with the pointer, allowing the shift in the average to be detectable.

Now, weak measurements are often considered in scenarios where the system of interest is post-selected. This means that, after the interaction with the device, the measured system continues its evolution until it is strongly measured and “collapses” in a state

Z |ϕi = ϕ(o)|oi do, (4.20)

which is assumed to be non-orthogonal to |ψi, i.e., hϕ|ψi 6= 0. When the post-selection is made, the small entanglement between the system and the device that conducted the weak

56 measurement is destroyed. In this case, the ﬁnal state of the device is

Observe that Z oψ(o)ϕ(o) hϕ|O|ψi hOi ≡ do = (4.22) w hϕ|ψi hϕ|ψi depends only on the measured system. This quantity, which was introduced in 1988 by Aharonov, Albert, and Vaidman in Ref. [121], is called the weak value (WV) of the operator O. It should be noticed that, diﬀerently from the expected value hOi, the WV is, in general, a complex number.

Now, assuming |g(hOiw − hOi)| ~ and recognizing

Z hϕ|ψi = ψ(o)ϕ(o) do, (4.23) it holds that

Therefore, the measurement device is purely aﬀected by the WV.

As an example, consider again the case where the initial states of the device and the system of

57 Figure 4.2: Effect of post-selections in weak measurements. The initial distribution of the measurement device (gray curve) is a Gaussian with zero mean and a standard deviation of σ = 0.4. The weak measurement slightly changes the distribution of the device (pink area). However, when considering post-selections of the measured system, the device’s distribution is affected. The purple curves show the (unnormalized) probability distributions after two different (and complementary) post-selections are made. interest are given by Eqs. (4.16) and (4.17), respectively. Then, if the qubit is post-selected in the state |0i, Eq. (4.18) implies that the (unnormalized) final probability distribution of the device is 2 1 − x−g |ξ|2(x) = e σ . (4.25) 6σ2π

This situation is represented in Fig. 4.2(a). Moreover, if the post-selection is made on the state |1i, the ﬁnal (unnormalized) distribution of the device is

2 1 − x+g |ξ|2(x) = e σ , (4.26) 3σ2π which is illustrated by Fig. 4.2(b).

As it can be observed, the distribution of the device is, in general, signiﬁcantly changed when considering weak measurements with post-selections. In fact, it can be shown from the deﬁnition in Eq. (4.22) that the WV can assume any complex value. When it has an imaginary part or its value goes outside the range of eigenvalues of the operator O, the WV is said to be anomalous.

To conclude, it should be mentioned that WVs arose from the study of the two-state vector

58 formalism of quantum mechanics, introduced by Aharonov, Bergmann, and Lebowitz [122] in 1964. This formalism considers that the time asymmetry existing in the dynamics of quantum systems due to the “collapse” of the wave function is the result of an incomplete setting of boundary conditions. Speciﬁcally, it considers that the state that can be attributed to a system in a certain instant of time does not depend only on how the system was prepared (the pre-selection), but also on the state it will be measured in the future (the post-selection). The idea is, then, that the post-selection travels backward in time and combines with the pre-selection, which travels forward in time, to deﬁne the “reality” of the system at the present [123–125].

Also, the research on WVs has become a rich area of study [126–142]. In particular, it has uncovered many surprising eﬀects in quantum theory [58, 143–147], besides shedding some light in the understanding of problems that were already known [148–152], like the interaction-free measurements [148, 153]. Also, anomalous WVs are proof of contextuality [154, 155]. Finally, WVs inspired new type of quantum experiments [156–163].

4.2 Quantization of the Sources of Electromagnetic Fields

Classical or quantum treatments of electromagnetic systems are characterized for being gauge invariant. It can be shown, however, that, when the source of electromagnetism is quantized, a more careful analysis of the problem is required. This section presents the model introduced in Ref. [164] and further studied in Ref. [165], which includes the quantization of the sources of the ﬁeld. Moreover, it discusses the delicate aspects of gauge transformations in this context.

Consider an inﬁnitely long cylindrical shell with radius a, rotational inertia Ic, and uniform charge density σ rotating around its long axis of symmetry (which coincides with the z axis) with angular velocityη ˙, in a similar manner as the solenoid represented in Fig. 2.1. Such a

59 cylinder generates a uniform magnetic ﬁeld on its interior given by

B~ = µ0σηa˙ z.ˆ (4.27)

Outside of it, the magnetic ﬁeld vanishes. Also, a uniformly charged wire is added in the z axis to cancel the electric ﬁeld on the exterior of the cylinder. With that, observe that this cylinder behaves like a solenoid. Finally, consider a charge q with mass µ traveling outside the cylinder in the xy plane, and let the coordinates of such a charge be given by (r, θ).

First, a classical treatment of the problem is given. For that, the joint system of the cylinder and the charge has its dynamics described by the Lagrangian

1 1 1 L = µr˙2 + I θ˙2 + I η˙2 + qτη˙θ,˙ (4.28) 2 2 d 2 c

2 where Iq = µr is the moment of inertia of the charge and τ is a constant associated with the interaction between the cylinder and the particle. From the Lagrangian, the canonical momentum with respect to each coordinate variable can be found. The canonical radial momentum of the particle is ∂L p ≡ = µr.˙ (4.29) r ∂r˙

Also, the canonical angular momentum of the charge is

∂L ˙ pθ ≡ = Iqθ + qτη.˙ (4.30) ∂θ˙

Furthermore, the canonical angular momentum of the cylinder is

∂L p ≡ = I η˙ + qτθ.˙ (4.31) η ∂η˙ c

60 Solving forη ˙ in Eq. (4.31) and replacing it in Eq. (4.30), it holds that

˙ pθ qτpη θ = 0 − 0 , (4.32) Iq IqIc

0 2 2 where Iq = Iq − q τ /Ic. Moreover,

pη qτpθ η˙ = − 0 . (4.33) Ic IqIc

The Hamiltonian of the system can, now, be derived with the Legendre transform. In fact, 2 2 neglecting terms with τ /Ic ,

2 ˙ 1 1 qτ 1 2 H = prr˙ + pθθ + pηη˙ − L ≈ pr + 0 pθ − pη + pη. (4.34) 2µ 2Iq Ic 2Ic

0 Now, take the limit Ic 1 such thatη ˙ ≈ τ/Ic is a ﬁnite constant K/2π. Then, Iq ≈ Iq and

2 1 1 qK 1 2 H ≈ pr + pθ − pη + pη. (4.35) 2µ 2Iq 2π 2Ic

With that, the system can be quantized. The Hamiltonian associated with it is

2 1 1 qK 1 2 H = Pr + Pθ − Pη + Pη , (4.36) 2µ 2Iq 2π 2Ic where Pr, Pθ and Pη are Hermitian operators canonical conjugated of the observable coordinates R, Θ and E, respectively. Observe that each operator P acts on a diﬀerent HS. Furthermore, the “operator vector potential” associated with the cylinder is

K A~ = R−1P θ,ˆ (4.37) 2π η i.e., it is given in the Coulomb gauge.

61 Now, there is a subtlety involved in gauge transformations that arises when the transformation is a function of non-commuting observables. To see that, consider the case where the total time derivative of the function f = −qτηθ˙ is added to the Lagrangian in Eq. (4.28). The new Lagrangian is, then,

1 1 1 L0 = µr˙2 + I θ˙2 + I η˙2 − qτηθ,¨ (4.38) 2 2 d 2 c

and it depends on the angular accelerationη ¨ of the cylinder. A rigorous derivation of the Hamiltonian associated with this Lagragian, and its respective quantization is presented in Ref. [165]. Here, it should just be noticed that this gauge chance corresponds to the application of the unitary transformation

U = e−iqKΘPη/2π~ (4.39)

2 2 2 to H in Eq. (4.36). Neglecting terms with q K Ic/4π and using the canonical commutation

relation between Θ and PΘ,

2 qK qK qK † 2 U Pθ − Pη ≈ Pθ − i PθΘPη ⇒ U Pθ − Pη U ≈ Pθ . (4.40) 2π 2π~ 2π

Moreover, this operation translates the coordinate η in such a way that it becomes η0 =

η − qKθpη/2π. Thus, similarly to what was done in Eq. (2.26), the new Hamiltonian is

0 1 2 1 2 1 2 H = UHU = Pr + Pθ + Pη0 (4.41) 2µ Iq 2Ic

0 since Pη0 = −i~∂/∂η is equivalent to Pη. Observe that, now, the coordinate of the cylinder is η0. It can, then, be veriﬁed that

0 qK E ,Pθ = −i I, (4.42) 2π~

62 where I is the identity operator that acts in the HS space associated with the coordinates η0 and θ. This means that the usual canonical commutations (which, in particular, consider that the cylinder’s and the charge’s observables commute with each other) do not hold anymore. As stated in Ref. [165], this has a physical interpretation: a measurement of E0 introduces an angular acceleration in the cylinder, which produces an electric ﬁeld outside of it that changes Pθ.

Therefore, although the Hamiltonian of any system can be quantized in an arbitrary gauge, the quantization process is more delicate when the sources of the electromagnetic ﬁeld are involved, and the Coulomb gauge seems to be the most appropriate choice.

4.3 Complex Vector Potentials

In the problem studied in the previous section, assume the charge travels in a circular 2 trajectory with radius r and the inner evolution of the cylinder, i.e., Pη /2Ic is negligible. In this case, the “operator vector potential” associated with the cylinder in the Coulomb gauge is K A~(r) = P θ.ˆ (4.43) 2πr η and the Hamiltonian in Eq. (4.36) can be written as

1 qK 2 H = Pθ − Pη . (4.44) 2Iq 2π

Also, denote by |mi the eigenstates of Pη such that

Pη|mi = m~|mi (4.45)

63 for every m ∈ Z. Now, if the charge encloses the cylinder in the superpositon

1 |ψ0i = √ (|ψLi + |ψRi) , (4.46) 2 where |ψLi represents a packet that passes to the left of the cylinder and |ψRi, a packet that travels to the right of it, as in Fig. 2.1, the state of the joint system after the particle encircles the cylinder can be written as

1 iqKm |Λi = √ |ψLi + e |ψRi ⊗ |mi. (4.47) 2

Thus, measurements of the position of the particle after the two wave packets recombine, closing the loop around the cylinder, will reveal the shift in the interference pattern characteristic of the AB eﬀect. In this case, the eﬀective vector potential has the standard classical form Km A~(r) = ~θ.ˆ (4.48) 2πr

However, in general, the angular momentum of the cylinder can be prepared in a superposition of eigenstates of Pη, i.e., in a state

X |ξi = cj|mji, (4.49) j∈Z where the constants c are such that P c 2 = 1. Then, after the charge ﬁnishes the j j∈Z j loop, the state of the joint system is

c e−iKqmj/2 X j iKqmj |Λi = √ (|ψLi + e |ψRi) ⊗ |mji. (4.50) 2 j∈Z

In other words, the charge and the angular momentum of the cylinder become entangled.

64 Then, averaging over the states of the cylinder, the ﬁnal state of the charge after it closes the loop around the cylinder is

Because of the entanglement, the interference between |ψLi and |ψRi is, in general, destroyed.

Now, a particular scenario of interest is the case where the charge interacts weakly with the ~ vector potential A, i.e., when |Kq(mj − hPηi)| ~ for every j ∈ Z such that cj 6= 0. In this case, X 2 iKqmj iKqhPηi/~ X 2 iKq(mj~−hPηi/~) cj e = e cj e j∈Z j∈Z (4.52) ≈ eiKqhPηi/~.

Hence, the ﬁnal state of the charge is

1 iKqhPηi/~ |ψ1i ≈ √ |ψLi + e |ψRi (4.53) 2 and the eﬀective vector potential in the Coulomb gauge is

K A~(r) = hP iθ.ˆ (4.54) 2πr η

As a result, under the assumptions that |Kq(mj − hPηi)| ~ for every j ∈ Z such that cj 6= 0, the quantum treatment of the cylinder eﬀectively produces the usual classical ﬁeld expected by the average over the quantum states of the cylinder.

65 Now, suppose that, after the charge encircles it, the cylinder is post-selected in a state

X |ϕi = dj|mji (4.55) j∈Z such that hϕ|ξi= 6 0. From Eq. (4.50), it holds that the ﬁnal state of the charge after the post-selection of the cylinder is

C X −iKqmj/2 iKqmj/2 |ψ2i = √ cjdj e |ψLi + e |ψRi , (4.56) 2 j∈Z

Kq hP i X ±iKqmj/2 ±iKqhPηi/2~ X η cjdje ≈ e cjdj 1 ± i mj − 2 ~ j∈Z j∈Z Kq ±iKqhPηi/2~ X = e cjdj 1 ± i hPηiw − hPηi (4.57) 2~ j∈Z

±iKqhPηiw/2~ X ≈ e cjdj. j∈Z

Then, the ﬁnal state of the charge after the post-selection of the angular momentum of the cylinder is given by D iKqhPηiw/~ |ψ2i = √ |ψLi + e |ψRi , (4.58) 2 where D is a normalization constant, which is necessary because, as mentioned in Section

4.1, hPηiw is, in general, a complex number. Also, the eﬀective vector potential in this case is K A~(r) = hP i θ.ˆ (4.59) 2πr η w

66 It can, then, be concluded that in a system with pre and post-selection, the eﬀective electromagnetic vector potential is, in general, complex-valued. However, it should be noticed that the assumed conditions imply that the imaginary part of the vector potential, if it exists, is relatively small.

Another interesting effect that can be observed is the continuous change of probability of finding the charge on the left or the right arm of the loop. In fact, assuming the loop is a circle centered at the cylinder, the “probability” of finding the particle on the left and the right arm after each wave packet traveled an angle θ can be computed as

eaθ p (θ) = (4.60) L eαθ + e−αθ and e−aθ p (θ) = , (4.61) R eαθ + e−αθ where a = KIm hPηiw /π~ is a real number. By probability here, it is meant the square of the magnitude of the amplitude. In other words, the amplitude of the charge’s state on each arm changes continuously in magnitude while it encircles the cylinder. This counterintuitive behavior results from the fact that, before the post-selection, there is a continuous increase in the entanglement between the charge and the cylinder while the particle encircles the cylinder — and, of course, this assumes that the charge is not observed in between the pre and post-selection of the cylinder’s angular momentum. Nevertheless, weak measurements of the position of the charge should reveal this surprising behavior.

4.4 Discussion

The results presented in this section shows that the vector potential can, in general, assume a complex value. If that is the case, the charge seems to behave in a somewhat counterintuitive way in the AB scenario, with a constant change in the magnitude of the amplitude on each

67 arm.

It should be noted that, in the limits considered in Sections 4.2 and 4.3, the charged cylinder can be taken to be a macroscopic object with the state of uncertain angular momentum given by Eq. (4.49). Also, if its rotational inertia Ic is suﬃciently high, it is possible to know both the cylinder’s position and angular velocity with a relatively high degree of certainty

(since its angular velocity is given by the ratio Pη/Ic). As a result, such a cylinder qualiﬁes as a classical object.

Now, it is also possible to post-select the cylinder in a state |ϕi such that the vector potential in Eq. (4.59) is complex-valued. In this scenario, the complex vector potential is also a classical object.

This seems to suggest a need for the generalization of the correspondence principle, which refers to the connection between classical and quantum systems. The Ehrenfest theorem is often seen as a mathematical basis for this principle. However, it accounts only for real expectation values playing a role in classical physics. In fact, post-selections of quantum systems, and the WVs associated with them, are typically not considered in the study of the quantum-to-classical transition.

As shown here, nonetheless, even for macroscopic objects, complex WVs provide a more detailed physical picture than the usual real expectation values. Furthermore, the present results suggest that classical quantities (in the sense of quantities associated with macroscopic objects) are, in general, complex-valued. Also, because complex WVs are proof of contextuality, this work suggests that contextuality may play a role in the quantum-to-classical transition.

Finally, it should be noted that, even though the example considered here presents a scenario where the vector potential is complex, the same general derivation should hold for other physical quantities of macroscopic objects, like position and momentum.

68 5 Fueter Variables on Banach Algebras

In this chapter, the main results from Ref. [92] are presented. Speciﬁcally, a notion of analyticity introduced by Fueter for functions of a quaternionic variable [166] is extended to the functions of a single variable that takes value in a Banach algebra (BA) A.

For that, Fueter variables (FVs) are deﬁned in Section 5.3. Continuing, the referred notion of derivative of A-valued functions is presented, followed by Fueter expansions and the Gleason problem, the theory of rational functions (RFs), and spaces of Fueter series, which include the Fock-Bargmann-Segal and the Drury-Arveson spaces. Before that, a short discussion on the extension of the notion of derivatives is presented.

5.1 Real Derivatives and Their Extension to Other Algebras

In real analysis, the derivative of a function f of a real variable x at a point x0 ∈ R is deﬁned as the limit df f(x) − f(x0) (x0) ≡ lim ∈ R (5.1) dx x→x0 x − x0 whenever it exists. A clear necessary, but not suﬃcient, condition for this limit to exist in 0 a certain open set ΩR ⊂ R is that the function belongs to the set C (ΩR) of continuous functions in ΩR. Moreover, if the derivative of the function exists for every point in ΩR and 1 0 it is continuous, the function is said to belong to a subset C (ΩR) of C (ΩR).

Because the derivative of a function is itself a function, the same reasoning applied to f can be iterated to df/dx and higher order derivatives. Then, in general, a function belongs to a k set C (ΩR), for some k ∈ N, if it admits at least k continuous derivatives. As an example k k+1 of a function that belongs to C (R) but does not belong C (R), consider the function

69 k+1 f(x) = |x| for k = 2n, where n ∈ N. Its k-th derivative, given by

dkf (x) = (k + 1)! |x|, (5.2) dxk

is continuous but not diﬀerentiable at x = 0.

∞ If a function admits inﬁnite continuous derivatives, i.e., if it belongs to C (ΩR), it is called ∞ a smooth function. Moreover, if a C (ΩR) function can be written as a convergent power

series in a neighborhood of x0 ∈ ΩR, the function is said to be analytic at x0.

These notions can be extended to functions f of a complex variable z. The direct analogous

of Eq. (5.1) for the deﬁnition of the derivative of f at a point z0 ∈ C is

df f(z) − f(z0) (z0) ≡ lim ∈ C. (5.3) dz z→z0 z − z0

Writing z = x + iy, z0 = x0 + iy0, and

f(z) = u(x, y) + iv(x, y), (5.4)

where x, y, x0, y0 ∈ R, u and v are real functions, and i is the imaginary complex unit, it can be shown that the limit in Eq. (5.3) exists if and only if u and v can be derived with respect to each variable at least once at (x0, y0) and

 ∂u ∂v  (x0, y0) = (x0, y0) ∂x ∂y , (5.5)  ∂u ∂v  ∂y (x0, y0) = −∂x(x0, y0)

which are known as the Cauchy-Riemann equations. Complex functions that are diﬀeren-

tiable at a point z0 ∈ ΩC ⊂ C are called holomorphic functions at z0. Also, a complex

function f is said to be analytic at a point z0 ∈ C ∈ ΩC if it can be written as a convergent

70 power series in a neighborhood of z0 ∈ ΩC. A remarkable theorem in complex analysis states a function is holomorphic if and only if it is analytic [167].

Although they are called Cauchy-Riemann equations, the expressions in Eq. (5.5) ﬁrst appeared in d’Alembert’s study of resistance of ﬂuids in 1752 [168]. After Euler had already connected those equations to analytic functions [167], Cauchy, in 1814, used the system in Eq. (5.5) to create his theory of functions. Later, in 1851, Riemann studied this theory in his Ph.D. dissertation [169].

While the extension of Eq. (5.1) to the complex case works, it fails for arbitrary algebras. For instance, consider the case of the quaternions. Because of the lack of commutativity, the extension of Eq. (5.1) can be written either as

−1 lim (f(q) − f(q0)) (q − q0) ∈ H (5.6) q→q0 or −1 lim (q − q0) (f(q) − f(q0)) ∈ H, (5.7) q→q0 where, now, f is a function of the quaternionic variable q and q0 ∈ H. However, in either of these cases, the derivative would only exist for linear functions, i.e., if a1, a2, b1, b2 ∈ H,

f(q) = a1q + b1 (5.8) or

f(q) = qa2 + b2, (5.9) respectively [170]. Therefore, if one wants to build a non-trivial analysis on the quaternions, other deﬁnitions of derivatives have to be considered.

In Ref. [166], Fueter studied quaternionic-valued functions and introduced non-commuting

71 hypercomplex variables that allowed power series expansion of hypercomplex functions. Such variables are now known as FVs. They are deﬁned in a more general context in Section 5.3, right after the introduction of the BAs that are considered here.

5.2 The Algebra

The central object of study in most of this chapter are functions

f : A → A, (5.10)

where A is an arbitrary BA over the ﬁeld K of the real or the complex numbers. First, however, some deﬁnitions and notation are introduced.

Let a1,..., an ∈ A \ K be the generators of A that differ from the identity element of K, i.e., from a0 ≡ 1. For simplicity, the number n of generators is assumed to be finite, but it could, in principle, be taken to be infinity for all computations presented here. However, a more careful analysis would be required because, for instance, the operator defined in Eq. (5.19) would become a differential operator on infinitely many variables.

In some algebras, all “directions” are given by the elements ak, k ∈ Zn+1 ≡ {0, 1, . . . , n}. If that is the case, there exist coeﬃcients cjkl ∈ K, j, k, l ∈ Zn+1 such that

n X ajak = cjklal (5.11) l=0

∗ for every j, k ∈ Zn+1 ≡ {1, . . . , n}. As an example, consider the quaternions. Denoting the imaginary units of the quaternions by i, j and k, the product of two elements (say, ij) is related to the third one (k in this case).

In other algebras, on the other hand, the product ajak might result in a new “direction.”

72 An example of such algebras is the real ternary algebra, which is generated by the number 1 and an element e that is not real (nor complex). However, e2 gives a new direction in such an algebra. The quaternions themselves can be seen as another example when the elements 1, i, and j are considered their generators. In this case, ij results in a new direction of the algebra.

Then, in order to include algebras A for which Eq. (5.11) does not necessarily hold, consider a set of t-uples I associated with all linearly independent directions of the algebra. Speciﬁcally, if, for a certain j and k, the product ajak results in a new direction, then (j, k) might be

an element of I and a(j,k) ≡ ajak. Note that, in some algebras, there might be multiple

ways to build I. For instance, if ajak = −akaj, as is the case of Cliﬀord algebras, either

(j, k) or (k, j) should be part of I, but not both. In such case, if (j, k) ∈ I, akaj = −a(j,k).

Moreover, the direction a0 = 1 is not included in I.

∗ For simplicity of notation, instead of referring to the set I directly, a map from it into Zm+1, where m is the cardinality of I, is used. Hence, instead of considering indexes that take ∗ value in I, they will take value in Zm+1. Moreover, to avoid confusion, we also denote the “independent directions” of A by ek instead of ak. For instance, if the quaternions are

generated by a0 = 1, a1 = i, and a2 = j, then, in the notation that will be used hereby, one

must consider e0 = a0, e1 = a1, e2 = a2, and e3 = a1a2.

m+1 Therefore, with those deﬁnitions, the algebra A can be identiﬁed with the space K in the following way

( ) m m+1 X K 'A = a = akek a0, a1, . . . , am ∈ K . (5.12) k=0

73 In particular, Eq. (5.11) can be rewritten as

m X ejek = (χ`)jke`, (5.13) `=0

(m+1)×(m+1) where the matrices χ` belong to K for every ` ∈ Km+1. The collection of these matrices will be called the characteristic operators of the algebra A since they encode all properties of the product of the algebra. Moreover, each matrix χ` can be seen as a metric tensor-like object associated with the `-th direction of A. In fact, observe that, for any a, b ∈ A, writing m X ab = c`e`, (5.14) `=0 Eq. (5.13) leads to m X c` = aj(χ`)jkbk ≡ aχ`b (5.15) j,k=1 for every ` ∈ Km+1.

Now, the BAs A of interest in this work are endowed with an involution, denoted by †, which has the following properties:

∀a, b ∈ A, (ab)† = b†a†;

† † 2 ∀k ∈ K ⊂ A, kk = k k = |k| .

Observe that, in general, aa† is not a real (nor a complex) number. In particular, aa† = a†a does not always hold.

Continuing, because A is a BA, it has a norm N for which

N(ab) ≤ N(a)N(b), ∀a, b ∈ A. (5.16)

Such a norm may or may not be induced by the involution †. Nevertheless, it is assumed

74 that N(a) = N(a†) for every a ∈ A.

Note that because K ⊂ A, N is also a norm in K. Therefore, without loss of generality, it can be presumed that N(1) = 1. Then, generally, N(k) = |k|, where |k| denotes the usual

norm of k in C, which coincides with the norm of a real number if K = R. Moreover, it follows that N(ka) = |k|N(a) (5.17)

for every a ∈ A.

Now, introduced basic notations and deﬁnitions in the algebra A, it is possible to use Eq. (5.12) to rewrite the functions f of the type given by Eq. (5.10) as

m+1 f : K → A. (5.18)

In particular, most of the focus here is on the subset of functions of the type given by

Eq. (5.18) that are K-analytic, i.e., functions that are real analytic if K = R or complex holomorphic if K = C.

5.3 A General Principle

The ideas introduced by Fueter for the quaternions [166] were adapted by Malonek for Cliﬀord algebras [171, 172] and by other authors in other settings [173, 174]. These ideas can be also extended to the BAs considered here. To start, it should be noticed that, because m+1 of the identiﬁcation between K and A presented in Eq. (5.12), it is possible to identify

f in Eq. (5.10) as an A-valued function of m + 1 K-valued variables vk, k ∈ Zm+1.

1 m+1 Now, consider functions f of class C in an open set Ω ⊂ K . Also, assume they lie in

75 the kernel of the Cauchy-Fueter (CF) operator, deﬁned as

m X D ≡ D0 + ekDk, (5.19) k=1

where ∂ Dj ≡ (5.20) ∂vj for every j ∈ Zm+1, i.e., they satisfy Df = 0. (5.21)

Such functions are called left D-hyperholomorphic or left monogenic [171]. The word left is used because one can also consider functions f that belongs to the kernel of D when it acts on f from the right. Here, because functions that satisfy Eq. (5.21) are the focus, the word left will be omitted, and the functions are just called D-hyperholomorphic or monogenic. However, observe that every result obtained from now on can be easily adapted for right D-hyperholomorphic functions.

m+1 Then, denoting v = (v0, v1, ··· , vm) ∈ K , it holds that

m df X ∂f (tv) = v (tv) dt k ∂v k=0 k m m ! X ∂f X ∂f = v (tv) − v e (tv) (5.22) k ∂v 0 k ∂v k=1 k k=1 k m X ∂f = (v − e v ) (tv) k k 0 ∂v k=1 k and, then, m Z 1 d X Z 1 ∂f f(v) − f(0) = f(tv)dt = (v − e v ) (tv)dt. (5.23) dt k k 0 ∂v 0 k=1 0 k

76 The functions ∗ ζk(v) = vk − ekv0, k ∈ Zm+1, (5.24) are called the FVs or total regular variables [175] associated with D.

More generally, for a ﬁxed w ∈ Ω,

m df X ∂f (tv + (1 − t)w) = (v − w ) (tv + (1 − t)w) dt k k ∂v k=0 k m (5.25) X ∂f = (ζ (v) − ζ (w)) (tv + (1 − t)w) k k ∂v k=1 k

holds for every v ∈ Ω. Therefore,

m Z 1 d X Z 1 ∂f f(v) − f(w) = f(tv)dt = (ζ (v) − ζ (w)) (tv + (1 − t)w)dt. (5.26) dt k k ∂v 0 k=1 0 k

Following Malonek’s approach for the Cliﬀord algebra in Ref. [171], consider now ζ = m (ζ1, ζ2, ··· , ζm) and let H be the set of all such vectors. Then, there is a one-to-one m+1 m correspondence between K and H , namely

m+1 m K ' H = {ζ = (ζ1, ··· , ζm) | ζk = vk − ekv0; v0, vk ∈ K} . (5.27)

Because of this correspondence, the notations f(v) and f(ζ) are used interchangeably. More-

over, ζ(w) will often be denoted as ξ ∈ H m.

Observe that, although H m ⊂ Am, H m is not a submodule of Am. In fact, cζ only belongs m m to H for an arbitrary ζ ∈ H if and only if c ∈ K. As a consequence, the product of two D-hyperholomorphic functions is not always D-hyperholomorphic. For instance, if

77 ∗ j, k ∈ Zm+1, it holds that

2 ζjζk = vjvk − ejv0vk − ekv0vj + ejekv0 (5.28) and, then, 2 D(ζjζk) = ej, ek v0, (5.29)

where ej, ek ≡ ejek−ekej is the commutator of ej and ek. Because, in general, ej, ek 6= 0, the right-hand side of Eq. (5.29) does not always vanish.

However, it follows from Eq. (5.29) that

D(ζjζk + ζkζj) = 0. (5.30)

In general, as it will be seen in the next section, symmetrized products of FVs and, in particular, powers of a single FV are D-hyperholomorphic.

Finally, the operator

Z 1 ∂f (Rk(ξ)f)(ζ) ≡ (Rk(w)f)(v) ≡ (tv + (1 − t)w)dt, (5.31) 0 ∂vk

where w ∈ Ω is a ﬁxed element, is called the backward-shift operator centered at w (or at ξ). Using this deﬁnition, Eq. (5.26) can be rewritten as

m X f(v) − f(w) = (ζk(v) − ζk(w))(Rk(w)f)(v). (5.32) k=1

Furthermore, observe that ∂f (Rk(w)f)(w) = (w). (5.33) ∂vk 2 Then, if f belongs to C (Ω), Rk(w)f(v) is D-hyperholomorphic. Generally, if f belongs to

78 p C (Ω) for some p ∈ N, the process given by Eq. (5.29) can be iterated p times, generating the so-called Fueter polynomials, which are the subject of study of Section 5.5. In particular, functions of class C∞(Ω) give origin to Fueter series, studied in Section 5.6.

5.4 Hyperholomorphicity of Functions from Km+1 into A

Following similar arguments of Ref. [171], it is now shown that functions of FVs admit a notion of a derivative that, in a sense, generalizes the real and the complex cases deﬁned in Eqs. (5.1) and (5.3).

First, to set the framework, endow Am with the Hermitian form

m X † [ζ, ξ]A ≡ ξkζk, (5.34) k=1 which is, in general, A-valued. Moreover, deﬁning

m ∗ hk = (0, ··· , 0, 1, 0 ··· , 0) ∈ H , k ∈ Zm+1, (5.35) and m h0 = − (e1, e2, ··· , em) ∈ H , (5.36) it follows that an arbitrary element ζ ∈ H m can be written as

m X ζ = vkhk. (5.37) k=0

Furthermore,

[ζ, hk]A = ζk (5.38)

79 ∗ for every k ∈ Zm+1, and

m m X 2 X [ζ, h0]A = −v0 ek − vkek. (5.39) k=1 k=1

m Clearly, the set {h } ∗ is a canonical basis for A . Then, motivated by it, the norm k k∈Zm+1 in Am is deﬁned as m !1/2 X 2 kζkAm ≡ N([ζ, hk]A) . (5.40) k=1

Now, let L (Am, A) be the set of all left A-linear operators from Am into A. Then, if L ∈ L , it holds that L(au + bv) = aL(u) + bL(v) (5.41) for every a, b ∈ A and u, v ∈ Am. In particular, it follows that L ∈ L (H m, A) is A-linear from the left if L(aζ + bξ) = aL(ζ) + bL(ξ) (5.42)

∗ and, moreover, there exist constants Ak ∈ A, k ∈ Zm+1, such that L is uniquely characterized by

L(ζ) = ζ1A1 + ··· + ζmAm. (5.43)

With that set, it is possible to introduce a generalization of real diﬀerentiation and complex holomorphicity in the present setting. An A-valued function f is said to be (left) hyperholo- m m morphic at a point ξ ∈ Ω ⊂ H if there exists a (left) A-linear map Lξ ∈ L (H , A) such that N f(ξ + ∆ζ) − f(ξ) − Lξ(∆ζ) lim = 0, (5.44) ∆ζ→0 k∆ζkAm

in which case Lξ is the (left) derivative of f at ξ. The function is also said to be hyperholo-

morphic in the open set Ω if there exists an Lξ in Eq. (5.44) for every ξ ∈ Ω. This notion

80 of derivative is called a Fr´echetderivative.

Observe that, if f is hyperholomorphic, then its derivative at ξ, i.e., the linear map Lξ in Eq. (5.44) is unique. In fact, Eqs. (5.43) and (5.44) imply that

f(ξ + ∆ζ) − f(ξ) = ∆ζ1A1 + ··· + ∆ζmAm + o(k∆ζkAm), (5.45) where o(k∆ζk m) lim A = 0. (5.46) ∆ζ→0 k∆ζkAm

Then, assuming that f is a K-analytic function, i.e.,

∂f ∂f ∂f f(ζ + ∆ζ) − f(ζ) = ∆f(ζ) = ∆v0 + ∆v1 + ··· + ∆vm + o(|∆v|), (5.47) ∂v0 ∂v1 ∂vm with o(|∆v|) lim = 0, (5.48) ∆v→0 |∆v| it is possible to show that f is hyperholomorphic if and only if it is D-hyperholomorphic. In fact, deﬁning v0 ≡ ζ0 and writing

∗ vk = ζ0ek + ζk k ∈ Zm+1, (5.49)

Eq. (5.47) leads to

m ! m X X ∆f(ζ) = ∆ζ0 D0f + ekDkf + ∆ζkDkf + o(k∆ζkAm) k=1 k=1 (5.50)

= ∆ζ0Df + [∆ζ, ∇v] f + o(k∆ζkAm),

Pm where ∇v ≡ k=1 Dkhk. Comparing Eq. (5.50) with Eq. (5.43), one concludes that f is hyperholomorphic if and only if Df = 0, i.e., if and only if f is D-hyperholomorphic.

81 5.5 Fueter Polynomials

It is already established that FVs are hyperholomorphic. In general, however, their pointwise product is not. Nevertheless, as it will be shown now, these variables generate hyperholomorphic polynomials, called Fueter polynomials, with the symmetrized product.

The symmetrized product of N elements a1, . . . , aN of an algebra A is deﬁned as

1 X a × a × · · · × a = a a ··· a , (5.51) 1 2 N N! σ(1) σ(2) σ(N) σ∈SN

where the sum is over the set SN of all permutations of N indexes. Observe that, in general, the product × is non-associative, i.e.,

(a × b) × c 6= a × (b × c). (5.52)

Moreover, if A is commutative, the symmetrized product reduces to the regular product.

Furthermore, as pointed out by Malonek in Ref. [172], for every k ∈ N,

X k! (a + ··· + a )k = aα, (5.53) 1 N α! N α∈N0 ; |α|=k

N α α1 αN where α = (α1, ··· , αN ) ∈ N0 , α! = α1!α2! ··· αN !, and a = a1 × · · · × aN .

With those deﬁnitions, the same method used in Refs. [166, 173, 176] can be applied to show

m αk ∗ that, if α = (α1, ··· , αm) ∈ N0 , the symmetrized product of the terms ζk , k ∈ Zm+1, simply denoted by ζα, is hyperholomorphic. To start, rewrite ζα as

m 1 X X X ζα = ζ ··· ζ ζ ζ ··· ζ , (5.54) |α|! σ(1) σ(k−1) u σ(k+1) σ(|α|) σ∈S u=1 ∗ |α| k∈Z|α|+1; σ(k)=u

82 where |α| denotes the sum of the components of α. Then,

m 1 X X X D(ζα) = e ζ ··· ζ ζ ··· ζ − |α|! u σ(1) σ(k−1) σ(k+1) σ(|α|) σ∈S u=1 ∗ |α| k∈Z|α|+1; σ(k)=u m (5.55) 1 X X X − ζ ··· ζ e ζ ··· ζ . |α|! σ(1) σ(k−1) u σ(k+1) σ(|α|) σ∈S u=1 ∗ |α| k∈Z|α|+1; σ(k)=u

Thus, because the sums in the previous equations over terms of the type

ζuζσ(1) ··· ζσ(k−1)ζσ(k+1) ··· ζσ(|α|) (5.56) equals the sums over

ζσ(1) ··· ζσ(k−1)ζuζσ(k+1) ··· ζσ(|α|), (5.57) it can be shown that α v0D(ζ ) = 0. (5.58)

α α Finally, because v0 can be taken to be non-zero, Eq. (5.58) implies that D(ζ ) = 0, i.e., ζ is hyperholomorphic.

As a consequence, polynomials of FVs with the symmetrized product, i.e., linear combinations of ζα with diﬀerent α’s (and A-valued coeﬃcients on the right-hand side of the powers) are also hyperholomorphic. These polynomials are referred to as Fueter polynomials.

To conclude, observe that, in a sense, Fueter polynomials appear naturally as an expansion of functions that admit many derivatives from repeated iterations of the backward-shift 2 operator. To see that, assume that f is at least of class C (Ω) and observe that Rk(ξ)Rj(ξ) = ∗ Rj(ξ)Rk(ξ) for every j, k ∈ Zm+1. Then, the application of Eq. (5.32) to (Rk(ξ)f)(ζ) leads

83 to

m X ∂f f(ζ) = f(ξ) + (ζ − ξ ) (ξ) k k ∂v k=1 k m 1 X + (ζ − ξ )(ζ − ξ ) + (ζ − ξ )(ζ − ξ ) (R (ξ)R (ξ)f)(ζ) 2 k k j j j j k k k j j,k=1 |α| m X α ∂ f X (5.59) = (ζ − ξ) α (ξ) + (ζk − ξk) × (ζj − ξj) (Rk(ξ)Rj(ξ)f)(ζ) m ∂v α∈N0 ; j,k=1 |α|<2 |α| X α ∂ f X α α = (ζ − ξ) α (ξ) + (ζ − ξ) (R(ξ) f)(ζ). m ∂v m α∈N0 ; α∈N0 ; |α|<2 |α|=2

m More generally, recalling that ζ(w) = ξ and deﬁning elements ιk ∈ N0 as

∗ (ιk)j ≡ δjk, j, k ∈ Zm+1, (5.60) observe that Z 1 α α ∂(ζ − ξ) Rk(ξ)(ζ − ξ) = (tv + (1 − t)w)dt 0 ∂vk Z 1 α−ιk = αk(ζ(tv + (1 − t)w) − ξ) dt 0 (5.61) Z 1 |α|−1 α−ιk = αkt (ζ − ξ) (v)dt 0 α = k (ζ − ξ)α−ιk |α| α for every α such that αk 6= 0. Also, Rk(ξ)(ζ − ξ) = 0 if αk = 0. Observe that Eq. (5.61) justiﬁes the name backward-shift operator given to Rk(ξ).

As a consequence, every function f of class Cp(Ω) can be written as

α |α| X (ζ − ξ) ∂ f X α α f(ζ) = α (ξ) + (ζ − ξ) (R(ξ) f)(ζ). (5.62) m |α|! ∂v m α∈N0 ; α∈N0 ; |α|

84 5.6 Fueter Series and the Gleason Problem

This section focuses on the study of hyperholomorphic functions, i.e., functions which are

K-analytic and belong to the kernel of the CF operator, given by Eq. (5.19). After that, the solution to a problem often referred to as the Gleason problem is studied.

To start, it is studied the convergence of inﬁnite iterations of R(ξ) on a K-analytic function f, i.e., the result of Eq. (5.62) in the limit p → ∞. Such a process leads, at least formally, to the Fueter series X α f(ζ) = (ζ − ξ) fα(ξ), (5.63) m α∈N0 where 1 ∂|α|f f (ξ) ≡ (ξ). (5.64) α |α|! ∂vα

Then, it needs to be shown that the series in Eq. (5.63) converges in an open neighborhood of ξ.

m+1 To do so, observe that, if a function f that takes value in K is K-analytic in a neighborhood Ω(w) of a point w, then the power series

X α f(v) = (v − w) Fα(w), (5.65) m+1 α∈N0

where 1 ∂|α|f F (w) ≡ (w), (5.66) α α! ∂vα

converges for every v ∈ Ω(w). In other words, there exist K, r1, ··· , rm > 0 such that

K K N(F ) ≤ ≡ . (5.67) α α0 αm α r0 ··· rm r

85 Observe that the coeﬃcients fα and F(0,α) can have their norm compared. In fact, using Eqs. (5.64) and (5.66), it can be derived that

N(fα) ≤ N(F(0,α)) (5.68)

m for every α ∈ N0 .

∗ Now, letting σ1, ··· , σm > 0 be such that σk < rk for all k ∈ Zm+1 and, again, denoting by m m+1 ξ the element in H that corresponds to the point w ∈ K , it is possible to deﬁne, in a similar way done in Ref. [172], the set

m ∗ m Uξ(σ) = ζ ∈ H N(ζk − ξk) ≤ σk, k ∈ Zm+1 ⊂ H . (5.69)

m+1 If a function f given by Eq. (5.65) is hyperholomorphic in a neighborhood of w ∈ K ,

the Fueter series given by Eq. (5.63) converges absolutely for every ζ ∈ Uξ(σ). To prove that, ﬁrst observe that, while either Eq. (5.63) and

∞ X X α f(ζ) = (ζ − ξ) fα(ξ) (5.70) m k=0 α∈N0 ; |α|=k

are, at least formally, possible representations of power series of functions of FVs, they generally have diﬀerent domains of convergence. This is the case, for instance, if A is a

Cliﬀord algebra [177]. Nevertheless, in Uξ(σ), the two expressions, if individually convergent, coincide. In fact, m α Y αk α α N((ζ − ξ) ) ≤ N(ζk − ξk) ≤ σ < r (5.71) k=1

86 and, as a consequence,

  X α X α X σ α N  (ζ − ξ) fα(ξ) ≤ σ N (fα(ξ)) ≤ K < ∞, (5.72) m m m r α∈N0 α∈N0 α∈N0 where Eqs. (5.67) and (5.68) were used. Hence, Eq. (5.63) converges absolutely in the domain Uξ(σ).

With the convergence of Fueter series of K-analytic functions proved, the Gleason problem can be considered. First, the Cauchy-Kovalevskaya (CK) product, also known as convolution, centered at ξ ∈ H m is deﬁned as

α β α+β (ζ − ξ) a ξ (ζ − ξ) b = (ζ − ξ) ab, (5.73)

m where α, β ∈ N0 and a, b ∈ A. This product can be extended to power series. In fact, remembering that ξ = ζ(w), let f be given by Eq. (5.63) and

X α g(ζ) = (ζ − ξ) gα. (5.74) m α∈N0

∗ Setting v0 = w0, power series in vk − wk, k ∈ Zm+1, for each function with coeﬃcients in A are obtained: X α f(w0, v1, . . . , vm) = (v − w) fα (5.75) m α∈N0 and X α g(w0, v1, . . . , vm) = (v − w) gα. (5.76) m α∈N0

Observe that, in those equations, the coeﬃcients fα and gα commute with the variables. Then, X α f(w0, v1, . . . , vm)g(w0, v1, . . . , vm) = (v − w) hα, (5.77) m α∈N0

87 where X hα ≡ fα−γgγ. (5.78) m γ∈N0 ; α−γ≥0 With that, the CK product of two power series is deﬁned as

X α (f ξ g)(ζ) ≡ (ζ − ξ) hα. (5.79) m α∈N0

n Moreover, if f(w) is invertible, there exist coeﬃcients dα ∈ A for every α ∈ N0 such that, m in a neighborhood of w in Ω ⊂ K , we have

−1 X α (f(w0, v1, . . . , vn)) = (v − w) dα. (5.80) n α∈N0

Then, the CK inverse of f can be deﬁned as

− X α f ξ (ζ) ≡ (ζ − ξ) dα. (5.81) m α∈N0

An important fact about the CK product that should be noted is that it dependents on the center ξ. For instance, consider, in the quaternionic setting, where e1 = i, e2 = j, and 2 e3 = k, the polynomials p1(ζ) = ζ1k and p2(ζ) = ζ1i. Then, if ξ = 0, P = p1 0 p2 = ζ1 j.

However, if ξ = i, because p1(ζ) = (ζ1 − i)k − j and p2(ζ) = (ζ1 − i)i − 1, the result of the CK product is 2 2 Q(ζ) = (p1 i p2)(ζ) = (ζ1 − i) j + j = ζ1 j − 2ζ1k (5.82) since ζ1 and i commute. Thus, P 6= Q, i.e., the CK products of p1 and p2 centered at the origin and centered at i are diﬀerent.

88 Observe that it is also possible to deﬁne the right-hand side CK product as

R X α (f ξ g)(z) ≡ hα(ζ − ξ) , (5.83) m α∈N0 where X α X α f(ζ) = fα(ζ − ξ) , g(ζ) = gα(ζ − ξ) (5.84) m m α∈N α∈N0 and the coeﬃcients hα are, once again, given by Eq. (5.78).

With that set, the Gleason problem can be ﬁnally presented. The problem was introduced by Gleason in his studies of maximum ideals of a commutative BA [178]. He considered n the algebra of holomorphic functions in the open unit ball in C that can be continuously extended to the boundary. Then, he proved that, if this algebra is ﬁnitely generated, the maximum ideals of functions that are zero at the origin is generated by the coordinate functions.

The problem of verifying if these ideals in algebras of holomorphic (and hyperholomorphic) functions are generated by the coordinate functions became known as the Gleason problem, and it was vastly studied in diﬀerent frameworks. A version of this problem in the present setting can be stated as a search for functions g1, ··· , gm such that a given hyperholomorphic function f with domain Uξ(σ) can be written as

m X f(ζ) − f(ξ) = (ζk − ξk) ξ gk(ζ) (5.85) k=1 for every ζ ∈ Uξ(σ).

Observe that, in principle, Eq. (5.32) is a solution to this problem if the CK product is replaced by the pointwise product. The problem, however, is not formulated in terms of the pointwise product because it has the disadvantage of, in general, not being hyperholomorphic.

89 Nevertheless, Eq. (5.32) can be seen as a good starting point in the obtaining of a solution to the problem. In fact, for every ζ such that v0 = w0, the CK product coincides with the pointwise product. As a result, the following equation also holds

m X f(ζ) − f(ξ) = (ζk − ξk) ξ (Rk(ξ)f)(ζ). (5.86) k=1

However, not all solutions to the Gleason problem are given by Eq. (5.86). To see that, let

G denote the space of functions f ∈ G for which there exist g1, g2, ··· , gm ∈ G that solve the Gleason problem. The space G is said to be resolvent invariant. Moreover, let R be the space of Rk(ξ)-invariant functions, called backward-shift invariant, i.e., the space for which gk = Rk(ξ)f. Then, it can be proved that R $ G.

First, it is necessary to characterize the elements of G. For that, it will be assumed that G is ﬁnite-dimensional. Also, let f ∈ G and G be a matrix-valued hyperholomorphic function whose columns form a basis for G. Then, by deﬁnition, there exist a constant column matrix

η with entries in A such that f = Gη. Moreover, there exist functions g1, . . . , gm ∈ G such that m X f(ζ) − f(ξ) = (ζk − ξk) ξ gk(ζ). (5.87) k=1 ∗ Finally, for every k ∈ Zm+1, there exist constant matrices Ak with entries in A such that gk = GAkη. Hence, Eq. (5.87) can be rewritten as

m X (G(ζ) − G(ξ)) η = G(ζ) ξ (ζk − ξk)Akη, (5.88) k=1 which implies that G is given by

m !− ξ X G(ζ) = G(ξ) ξ I − (ζk − ξk)Ak . (5.89) k=1

90 Now, letting f = Gη, where η is a constant column matrix with entries in A, it holds that

m !− ξ X f(ζ) − f(ξ) = G(ξ) ξ I − (ζk − ξk)Ak η − G(ξ)η k=1 m !! X = G(ζ) ξ I − I − (ζk − ξk)Ak η (5.90) k=1 m X = (ζk − ξk) ξ GAkη, k=1 i.e., there exist functions gk = GAkη ∈ G which solves the Gleason problem for f.

Therefore, a function f belongs to a ﬁnite-dimensional resolvent invariant space G if and only if it can be spanned by the columns of a matrix-valued function of the type given by Eq. (5.89).

Finally, to see that solutions given by Eq. (5.86) are only a subset of G, observe that, because a function f in R also belongs to G, f can be written as f = Gη, where G is of the form given by Eq. (5.89) and η is a constant column matrix with entries in A. Moreover, since gk = GAkη Rk(ξ)f ∈ G, as just seen, it holds that Rk(ξ)G = GAk. Furthermore, because the operators Rk(ξ) commute, Rk(ξ)Rj(ξ)G = Rj(ξ)Rk(ξ)G. Then, AkAj = AjAk for ∗ every j, k ∈ Zm+1.

This shows that functions in R are a particular type of functions in G for which all matrices

Ak commute. In fact, every function in G for which the matrices Ak commute belong to R.

To prove it, let f ∈ G be such that all matrices Ak in Eq. (5.89) commute. Let, moreover, η be a constant column matrix with entries in A such that f = Gη. Then, observe that the commutativity of the matrices Ak allows Eq. (5.89) to be rewritten as

X |α|! G(ζ) = (ζ − ξ)αG(ξ)Aα, (5.91) m α! α∈N0

91 α α1 α2 αm where A = A1 A2 ··· Am . Moreover, using Eq. (5.61),

(|α| − 1)! X α−ιk α (Rk(ξ)f)(ζ) = (ζ − ξ) G(ξ)A η m (α − ιk)! α∈N0 ; α≥ιk (|α| − 1)! X α−ιk α−ιk = (ζ − ξ) G(ξ)A Akη m (α − ιk)! (5.92) α∈N0 ; α≥ιk

= G(ζ)Akη

= gk(ζ).

In other words, a function belongs to R if and only if it is an element of G for which all matrices Ak in Eq. (5.89) commute.

Before concluding, note that, if f ∈ G, there exists a constant column matrix η with entries in A such that f = Gη, where G is given by Eq. (5.89). Moreover, f admits solutions to the

Gleason problem, which are given by gk = GAkη. Then,

m X f(ζ) = f(ξ) + G(ζ) ξ (ζk − ξk)Akη k=1 (5.93) m !− ξ m X X = f(ξ) + G(ξ) ξ I − (ζk − ξk)Ak ξ (ζk − ξk)Akη. k=1 k=1

This equation characterizes a hyperholomorphic RF, the topic of the next section.

5.7 Hyperholomorphic Rational Functions

RFs, i.e., quotients of polynomials, of FVs are studied now. These functions play an important role in many areas, from mathematical analysis to signal processing. For instance, because polynomials can be easily evaluated, RFs are widely used for approximation and interpolation of functions in numerical analysis. Also, in the complex setting, these functions

92 appear in the study of linear systems as the z-transform (or the Laplace transform) of the impulse response of time-invariant systems is a RF.

The focus here is on RFs which are analytic in a neighborhood of a ﬁxed ξ ∈ H m. From the theory of linear systems, it is known that such RFs can be written as

m !− ξ m ! X X R(ζ) = D + C ξ I − (ζk − ξk)Ak ξ (ζk − ξk)Bk , (5.94) k=1 k=1

∗ where Ak,Bk,C and D, for every k ∈ Zm+1, are matrices of appropriate sizes with entries in A. Eq. (5.94) is called a realization of R, a notion that originated with the theory of linear systems [179].

While Eq. (5.94) is clearly a ratio, with respect to the CK product, of Fueter polynomials, it is still necessary to prove the converse, i.e., to show that every RF analytic at ξ admits a realization of the type given by Eq. (5.94). For that, observe that the inverse of a function that is invertible at ξ and whose realization is given by Eq. (5.94), with D invertible, admits a realization of the form

m !− ξ m ! − ξ −1 −1 X X −1 R(ζ) = D −D C ξ I − (ζk − ξk)Ak ξ (ζk − ξk)Bk D , (5.95) k=1 k=1 where

−1 Ak = Ak − BkD C. (5.96)

This can proved by writing

! m m X −1 X (ζk − ξk)Bk D C = (ζk − ξk) Ak − Ak k=1 k=1 m ! m ! (5.97) X X = I − (ζk − ξk)Ak − I − (ζk − ξk)Ak k=1 k=1

93 and verifying that, indeed, −1 R(ζ) ξ R(ζ) = I. (5.98)

Now, if

m !− ξ m ! X X Ru(ζ) = Du + Cu ξ INu − (ζk − ξk)(Au)k ξ (ζk − ξk)(Bu)k , (5.99) k=1 k=1

∗ for u = 1, 2, gives two realizations of RFs with compatible sizes, then, for k ∈ Zm+1, it can be veriﬁed by direct computation that their product admits a realization given by Eq. (5.94) with     (A ) (B ) C (B ) D  1 k 1 k 2  1 k 2 Ak =   ,Bk =   , 0 (A ) (B ) 2 k 2 k (5.100) C = C1 D1C2 ,D = D1D2.

Moreover, their sum admits the realization

    (A ) 0 (B )  1 k   1 k Ak =   ,Bk =   , 0 (A ) (B ) 2 k 2 k (5.101) C = C1 C2 ,D = D1 + D2.

Thus, to show that any Fueter polynomial admits a realization, it suﬃces to prove that ∗ constant terms and terms of the form (ζk − ξk)M, for every k ∈ Zm+1, admit realizations. However, this is clear because

a constant term corresponds to the realization where Aj = B = C = 0, for every ∗ j ∈ Zm+1;

the function (ζk − ξk)M corresponds to C = M, Aj = D = 0, Bj = δjkIN , for every

94 ∗ j ∈ Zm+1.

Therefore, because every realization is a RF, as discussed, it can be said that a function R of FVs analytic at ξ is rational if and only if it admits a realization, which is given by Eq. (5.94).

Finally, to conclude, it can be shown that a function R of FVs analytic at the origin is rational if and only if there exists a ﬁnite-dimensional resolvent-invariant space G such that the Gleason problem is solvable for every f = Rη ∈ G, where η is a constant column matrix with entries in A. In fact, one direction of this statement was already proven in the discussion of Eq. (5.90), where it was shown that every f in a ﬁnite-dimensional G admits a realization. For the converse, assume that R is given by Eq. (5.94) and that there exists η such that f = Rη. Then,

m !− ξ m ! X X f(ζ) − f(ξ) = C ξ I − (ζk − ξk)Ak ξ (ζk − ξk)Bk η. (5.102) k=1 k=1

As a result, m !− ξ X gk = C ξ I − (ζk − ξk)Ak Bkη (5.103) k=1 solves the Gleason problem in a space G generated by the columns of a function given by Eq. (5.89).

5.8 Banach Modules of Fueter Series

A fundamental aspect of functional analysis and operator theory is the study of positive kernels and their associated reproducing kernel Hilbert spaces (HSs) of power series. However, because the present work considers power series in BAs, the constructions that typically lead to HSs in many settings give rise to Banach modules of Fueter series (BMFS). These

95 modules are studied now.

For convenience, only Fueter series centered at the origin are considered hereby. However, one can easily reproduce the results that are presented here for Fueter series centered at a diﬀerent point ξ. Moreover, because of this choice, the CK product is simply denoted by

(instead of 0) and the backward-shift operator, by Rk (instead of Rk(0)).

To set the framework, let c = (c ) m be a family of non-null real numbers. Also, let α α∈N0 ◦ P α W (c) denote the module of functions f(ζ) = m ζ fα with coeﬃcients fα ∈ A such α∈N0 that  1/2 X 2 kfk ≡  |cα|(N(fα))  < ∞. (5.104) m α∈N0

◦ Observe that Eq. (5.104) deﬁnes a norm in W(c). In fact, for every a ∈ A,

 1/2 X 2 kafk =  |cα|(N(afα))  ≤ N(a)kfk. (5.105) m α∈N0

However, if a is restricted to K, i.e., if a ∈ K ⊂ A,

kafk = |a|kfk, (5.106) where Eq. (5.17) was used. Moreover,

 1/2 X 2 kfk = 0 ⇒  |cα|(N(fα))  = 0 m α∈N0 (5.107) m ⇒ fα = 0, ∀α ∈ N0 ⇒ f = 0.

96 ◦ P α Finally, if g(ζ) = m ζ gα belongs to W(c), α∈N0

2 X 2 kf + gk = |cα|(N(fα + gα)) m α∈N0 X 2 ≤ |cα| [N(fα) + N(gα)] m α∈N0 X h 2 2i (5.108) ≤ |cα| N(fα) + 2N(fα)N(gα) + N(gα) m α∈N0 ≤ kfk2 + 2kfk · kgk + kgk2

≤ (kfk + kgk)2 .

◦ P α P α Now, let f(ζ) = m ζ fα and g(ζ) = m ζ gα be two elements of W (c). Then, α∈N0 α∈N0 the Hermitian form X † [f, g] ≡ cαgαfα (5.109) m α∈N0 converges in A. In fact, using the Cauchy-Schwartz inequality,

X N([f, g]) ≤ |cα|N(fα)N(gα) m α∈N0  1/2  1/2 X 2 X 2 (5.110) ≤  |cα|N(fα)   |cα|N(gα)  m m α∈N0 α∈N0 ≤ kfk · kgk.

In cases where the algebra A does not have zero divisors, the results presented by Paschke in Ref. [180] are, then, valid in the context of the present section because the form deﬁned in Eq. (5.109) satisﬁes the conditions to be what is called an A-valued inner product. However, the goal here is the study of a more generic scenario, where A is allowed to have zero divisors.

97 ◦ With the Hermitian form deﬁned in Eq. (5.109), W(c) admits the reproducing kernel

X ζα(ξα)† Kc(ζ, ξ) ≡ (5.111) m cα α∈N0

in an open neighborhood of the origin in H m deﬁned by

  2α  m X (N(ζ))  Ω(c) = ζ ∈ H < ∞ (5.112)  m |cα|  α∈N0 since α X † ξ † [f(·),Kc(·, ξ)b] = cα b fα = b f(ξ). (5.113) m cα α∈N0

◦ Now, it is important to note that, a priori, the module W (c) is not complete. However, by a general theorem on metric spaces, it has a completion to a Banach space, which is unique up to a metric space isometry – see, e.g., pages 54-58 of Ref. [181]. To see that, consider the ◦ vector module of Cauchy sequences in W (c), and say that two of such sequences, say (fn)

and (gn), are equivalent if

lim kfn − gnk = 0. (5.114) n→∞

It can be checked that this is indeed an equivalence relation. The quotient module associated with it is denoted by CS. Like in the complex case, the formula

∼ k f k = lim kfnk (5.115) n→∞

∼ where (fn) belongs to the equivalence class f , does not depend on the given representative in the equivalence class, and deﬁnes a norm on CS. Then, it follows from Eqs. (5.109) and

98 (5.113) that, for f ∈ CS, the limit

† † b f(w) ≡ lim b fn(w) (5.116) n→∞ exists in the topology of A and is independent of the given representative of the equivalence class.

Finally, the representation of CS as a module of functions is denoted by W(c). It can veriﬁed that the space W(c) endowed with the norm given by Eq. (5.115) is a Banach module, in ◦ which W (c) is naturally embedded in a dense way. Moreover, Eq. (5.113) also holds in W(c).

If for a certain algebra the form given by Eq. (5.109) is such that [f, f] ≥ 0, i.e., it is a positive real number, for every f in W(c), then W(c) is a Hilbert module. If, on the other hand, [f, f] is a real number for every f ∈ W(c) but it is not always positive for every f, then W(c) is a Potryagin or a Krein module. For example, the Banach module of power series associated with modules with positive coeﬃcients cα when the algebra A is the quaternionic algebra is a HS. In the case of split-quaternions, however, the analogous module is a Pontryagin module.

This discussion can be extended by writing

m   X † X X † [f, g] = cαgαfα =  cαgαχ`fα e`, (5.117) m m α∈N0 `=0 α∈N0 where the characteristic operators of the algebra A, deﬁned in Eq. (5.15), have been used.

Then, the inner summation can be identiﬁed with K-valued inner product structures in

99 certain spaces (denoted by K`(c)) with the following deﬁnition

X hf, gi ≡ c g† χ f . (5.118) K`(c) α α ` α m α∈N0

Observe that each K`(c) constitutes, in general, a Hilbert or a Pontryagin space. What determines if K`(c) ends up being a Hilbert or a Pontryagin space is the coeﬃcients cα together with the characteristic operator χ`. For more information on Pontryagin or Krein spaces, see, e.g., Refs. [182–184].

An important characteristic of W(c) when it is endowed with the Hermitian form given by Eq. (5.109) is that, if g ∈ W(c) is such that

[f, g] = 0 (5.119)

m for every f ∈ W(c), then g = 0. In fact, for every α ∈ N0 ,

α † [ζ , g] = cαgα. (5.120)

Because cα ∈ R is assumed to be non-zero, Eq. (5.119) implies that gα = 0 for every α, i.e., g = 0. Similarly, it can be shown that, if f is ﬁxed and Eq. (5.119) holds for every g ∈ W(c), then f = 0.

Now that the module W(c) is introduced, the rest of this section focus on the study of operators that act on it. First, let O be an operator in W(c) and assume that it admits an adjoint, i.e., there exists an operator A characterized by

[Af, g] ≡ [f, Og] (5.121) for every f, g ∈ W(c) for which the term on the right-hand side converges. Then, A is

100 unique. In fact, suppose O admitted two adjoints, say A1 and A2. In this case,

[A1f − A2f, g] = [A1f, g] − [A2f, g] = [f, Og] − [f, Og] = 0. (5.122)

Therefore, following the discussion of Eq. (5.119), it can be concluded that

A1f − A2f = 0 (5.123)

for every f ∈ W(c), which implies that A1 = A2. Because the adjoint of an operator O is unique, it will be denoted by A = O∗.

To address the question of whether an operator admits an adjoint in the ﬁrst place, suppose m+1 O is bounded in K`(c) for every ` ∈ Z . Then, it is a result from complex analysis that, in each K`(c), O admits an adjoint, i.e., there exists an operator A` such that

hOf, gi = hf, A gi . (5.124) K`(c) ` K`(c)

To show that the existence of such adjoints in every K`(c) implies the existence of an adjoint in W(c), observe that hOf, gi = [χ Of, g] (5.125) K`(c) ` and hf, A gi = [χ f, A g] . (5.126) ` K`(c) ` `

These equations, together with Eq. (5.124), lead to

∗ [χ`Of, g] = f, χ` A`g , (5.127)

101 which, in turn, implies that

" m ! # X † ∗ [Of, g] = f, e`χ` A` g . (5.128) k=0

Then, using the deﬁnition of the adjoint operator, it can be concluded that

m ∗ X † ∗ O = e`χ` A`. (5.129) k=0

Therefore, it is proved that, if the operator O is bounded in every K`(c), it admits an adjoint in W(c).

Before present speciﬁc examples of BMFS, the multiplication operator, which plays an important role in those modules, will be deﬁned. For that, let M , for every k ∈ ∗ , ζk Zm+1 denote the CK multiplication operator by ζk, i.e., if f belongs to W(c), then

M f ≡ ζ f. (5.130) ζk k

More generally, if W(c) and W(d) are two BMFS, a multiplier Ms can be deﬁned

Ms : W(c) → W(d) (5.131) Msf 7→ s f, where f ∈ W(c) and s f ∈ W(d).

5.8.1 Fock-Bargmann-Segal Module

The Fock-Bargmann-Segal (FBS) module is the particular Banach module of Fueter series W(c) for which

cα = α!. (5.132)

102 Its reproducing kernel is, then, given by

X 1 α α † Kc(ζ, ξ) = ζ (ξ ) (5.133) m α! α∈N0

and it is endowed with the Hermitian form

X † [f, g] = α! gαfα. (5.134) m α∈N0

One of the results that most characterizes the FBS module, at least in the algebras where it was already studied, is the fact that the adjoint of the multiplication operator is the derivative operator on it. In the present framework, if the derivative operator ∂k is taken to be

X α−ιk ∂kf ≡ αkζ fα (5.135) m α∈N0 ∗ for every k ∈ Zm+1, this result also holds, i.e.,

[∂ f, g] = f, M g (5.136) k ζk

m for every f, g ∈ W(c). To prove that, let α, β ∈ N0 . Then,

h i h i α β α−ιk β ∂kζ , ζ = αkζ , ζ

= (α − ιk)! αk δα−ι ,β k (5.137) = α! δ α,β+ιk h i = ζα, ζβ+ιk .

103 5.8.2 Drury-Arveson Module

The Drury-Arveson (DA) module, also known as the symmetric Fock module, is the partic-

ular case of a Banach module of Fueter series which has its coeﬃcients cα given by

α! c = , (5.138) α |α|!

i.e., the module W(c) with reproducing kernel

X |α|! α α † Kc(ζ, ξ) = ζ (ξ ) (5.139) m α! α∈N0 and Hermitian form X α! † [f, g] = gαfα. (5.140) m |α|! α∈N0

Because contractions are important players in the study of DA modules, the idea of a contractive operator needs to be introduced in the present framework. First, observe that, in A, a “natural” notion of non-negativity is given by

aa† 0, (5.141)

for every a ∈ A. Note that such a deﬁnition does not always imply real non-negativity — even if aa† is a real number. For example, aa† can be a negative real number in the setting of the split-quaternions.

With Eq. (5.141), an operator O in W(c) is said to be a contraction if

[Of, Of] [f, f] (5.142)

for every f ∈ W(c).

104 One of the common characteristics of the DA modules already studied in the literature is the fact that the multiplication operators are a contraction from the module into itself. This is also the case here since, for every f ∈ W(c), it holds that

X M f, M f = ζα+ιk f , ζα+ιk f ζk ζk α α m α∈N0 X (α + ιk)! † = fαfα (5.143) m (|α| + 1)! α∈N0 X αk + 1 † = cαfαfα . m |α| + 1 α∈N0

Then, letting s c (|α| − α ) d = α k , (5.144) α |α| + 1 it can be concluded that

X |α| − α [f, f] − M f, M f = k c f†f ζk ζk α α α m |α| + 1 α∈N0 X † = (dαfα) (dαfα) (5.145) m α∈N0 0.

Another important characteristic of DA modules is the fact that the multiplication and the backward-shift operators are adjoints of each other. Again, this holds in the present setting, as can be seen in a proof that is similar to the one presented for the quaternions in Ref. m [185]. In fact, for every f, g ∈ W(c) and α, β ∈ N0 such that αk ≥ 1,

h i α (β + ι )! h i h i R ζα, ζβ = k ζα−ιk , ζβ = k δ = ζα, ζβ+ιk = ζα, M ζβ (5.146) k |α| (|β| + 1)! α−ιk,β ζk

105 ∗ for every k ∈ Zm+1. Finally, if β is such that βk = 0,

h i h i h i R ζα, ζβ = 0 = ζα, ζβ+ιk = ζα, M ζβ . (5.147) k ζk

Before concluding, an important tool for interpolations in complex DA spaces, namely the m Blaschke factor, is introduced. For every ξ such that kξkAm < 1, where the norm in A is deﬁned in Eq. (5.40), the Blaschke factor Bξ is deﬁned as

∗ 1/2 ∗ − ∗ −1/2 Bξ ≡ (1 − ξξ ) (1 − ζξ ) (ζ − ξ)(I − ξ ξ) , (5.148) where  †  ξ1    †   ξ  ∗  2  ξ ≡  .  (5.149)  .     †  ξm is the transpose of ξ = (ξ1 ξ2 ··· ξm).

Now, let C be the operator of evaluation at the origin, i.e., Cf = f(0) for every f ∈ W(c). Thus, it holds that X I − M M∗ = C∗C. (5.150) ζk ζk ∗ k∈Zm+1 With that, the following identity holds in the DA module W(c):

∗ ∗ 1/2 ∗−1 ∗ ∗−∗ ∗ 1/2 I − BξBξ = (1 − ξξ ) I − MζMξ C C I − MζMξ (1 − ξξ ) , (5.151)

where Bξ is the operator of multiplication by the Blaschke factor. The proof follows the ∗ one presented in Refs. [186] and [185]. First, observe that the operators I − MξMξ and ∗ I − MξMξ are self-adjoint. Moreover, because kξkAm < 1, they are also contractive. As

106 a result, their square root (and the inverse of their square root) exist. Then, deﬁning the

Halmos extension of −Mξ, as done in Ref. [187],

 −1/2 −1/2  I − M M∗ −M I − M∗M  ξ ξ ξ ξ ξ  H ≡ −1/2 −1/2 (5.152)  ∗ ∗ ∗  −Mξ I − MξMξ I − MξMξ and setting   I 0  W(c)  J =   , (5.153) 0 −IW(c)m it holds that HJH∗ = H∗JH = J. (5.154)

Then, using Eq. (5.150),

∗ ∗ C C = I − MζMζ   I   = I Mζ J   Mζ   I ∗   = I Mζ HJH   (5.155) Mζ   ∗ X1 = (X1 X2) J    ∗  X2

∗ ∗ = X1X1 − X2X2 , where ∗ ∗−1/2 X1 = I − MζMξ I − MξMξ (5.156) and ∗ −1/2 X2 = Mζ − Mξ I − MξMξ . (5.157)

107 5.9 Discussion

This chapter started with a short explanation of the problem of extending the notion of derivatives to functions beyond functions of a complex variable. Nevertheless, following an approach started by Fueter for functions of a quaternionic variable and generalized by Malonek for functions of Cliﬀord algebras, a derivative of functions of a single variable that takes values in a BA A was constructed. This notion required the introduction of “special directions” in A: the FVs. With those variables, hyperholomorphic polynomials and series were constructed with the use of the symmetrized and the CK product.

An aspect of the CK product introduced here in a way that seems to never been done in previous works and, then, requires further investigation is the fact that this product was

defined at an arbitrary center ξ ∈ H m. Since the coefficients of the product of polynomials and power series are defined from the center of this product, it can be said that the point

ξ determines the origin of the extension of the domain of functions from K to A. Then, because, in general, different centers are associated with different functions, the extension associated with them generates different mathematical structures. It would be interesting to better understand these structures and to look for possible relations between them.

Besides that, some applications of Fueter series were also considered. Basic deﬁnitions and results on the theory of RFs and realizations and BMFS were presented. These subjects are fundamental players in the study of functional analysis, operator theory, and signal processing. Because of it, in the complex setting, they are well-developed ﬁelds. Then, it would be interesting to verify if some form of the known results from complex analysis holds in the setting presented here.

Moreover, the spaces K`(c) and their inner product were introduced in Section 5.8. It seems that a variety of characteristics of BMFS can be studied through them. This is a direction

that should be further explored. Because the spaces K`(c) take value in K, they might

108 allow the “transference” of many results from the real and from the complex settings to the framework introduced here.

Also, some examples of BMFS were given, namely the FBS and the DA modules. Again, the two modules were already extensively studied in other settings, which means that there exists a large number of results whose validity in the present framework needs to be verified. For instance, in the complex case, the Blaschke factor and the correspondent counterpart of Eq. (5.151) allow the study of interpolation problems in the DA space [186]. In other settings, this line of research leads to unexpected conclusions. For instance, in the quaternionic case, even the simplest interpolation problem of finding all functions f in the quaternionic DA module such that f(a) = 0 involves an infinite number of interpolation conditions of the form [185] (f ζα)(a) = 0 (5.158)

3 when α runs through N0. Similar problems also arise in the present setting and will be addressed elsewhere.

Finally, it should be noticed that, although Fueter’s approach was extended here, this is not the only possible notion that can be generalized from the complex case. For instance, in 2007, Gentili and Struppa introduced the idea of slice derivatives in the quaternionic setting [188]. The advantage of their approach is the fact that the resultant power series does not correspond to power series of special variables. Instead, they consist of powers of a variable q ∈ H. Similar ideas may hold in the setting presented here if more restrictions are imposed on it. This is another interesting research topic to be pursued in future works.

109 6 Analysis in Generalized Grassmann Algebras

This chapter presents the main results on analysis in closures of the Grassman algebra introduced in Refs. [93, 94]. The Grassmann algebra (GA) was first formalized by Grassmann in 1844 in a work that presented a formal framework for aspects of Cayley and Sylvester’s theory of multivectors [189]. Analysis in this algebra only started in 1899 with Cartan showing that the exterior algebra can be represented by it if the idea of derivatives by its generators is introduced [190]. The first application of this algebra in physics appeared in a work published by Martin in 1959, where the elements of the algebra were used in the study of “classical versions” of physical functions for fermions and the obtainment of their quantization through path integrals [191, 192]. Seven years later, this idea would be used by Schwinger to extend his quantum field theory to fermions [193]. Moreover, in 1979, Berezin independently started an extensive study of what is now known as supermathematics [194, 195].

The chapter is divided into two parts. In the ﬁrst one, which goes from Section 6.2 to Section 6.9, the closure of the GA with respect to the 1-norm is studied. In this setting, many important ideas from functional analysis are introduced, like the one-step extension of Toeplitz matrices, rational functions (RFs), Schur functions, reproducing kernels, and interpolation problems. In the second part, which starts in Section 6.10 and ends in Section 6.12, a class of stochastic processes is introduced for functions whose image lies in the closure of the GA with respect to the 2-norm. Also, a framework is introduced where the derivative of the stochastic variables can be studied as continuous functions. Finally, a discussion of the chapter and an indication of future research directions are presented in the last section. Before that, in the next section, basic results, deﬁnitions, and notations are introduced.

110 6.1 Grassmann Algebra

The GA Λ is deﬁned as the unital algebra over a ﬁeld K generated by 1 and a countable set

of elements in, which do not belong to K, are linearly independent over K, and satisfy

inim + imin = 0, (6.1)

where n, m = 1, 2,.... While Eq. (6.1) holds for a variety of hypercomplex numbers [196], the particularity of the generators of Λ is that Eq. (6.1) also holds for n = m, i.e.,

2 in = 0. (6.2)

Then, Λ has divisors of zero. Since K is often taken to be the ﬁeld of the complex numbers, it is assumed here that K = C.

The GA plays a fundamental role in supersymmetry and in quantum ﬁeld theory, where it allows the construction of path integrals for fermions [197]. Because of its importance to supersymmetry, an element of Λ is commonly referred to as a supernumber.

In this work, if the GA has N generators in, it will be denoted by ΛN . Moreover, the union of the algebras ΛN will be denoted by Λ, i.e., Λ ≡ ∪N∈NΛN , as illustrated in Fig. 6.1. After this introductory section, closures of Λ, where the algebra has an inﬁnite number of generators, will be the main focus of the chapter.

t To set the framework, I is deﬁned to be the set of t-uples (a1, . . . , at) ∈ N , where t runs through N and a1 < a2 < ··· < at. Whit that, iα ≡ ia1 ··· iat for any α = (a1, . . . , at) ∈ I. Then, an element z ∈ Λ can be written as the ﬁnite sum

X z = z0 + zαiα, (6.3) α∈I

111 Figure 6.1: Schematic representation of the increasing family of Grassmann algebras.

where the coeﬃcients z0 and za1,...,ak are complex numbers.

The term that does not contain any Grassmann generator, z0, is called the body of the number, and it is often denoted by zB. Meanwhile, zS = z − zB is said to be the soul of the number [198]. As it will be discussed later in this section, it is possible to give a meaning to the sum in Eq. (6.3) when it has an inﬁnite number of terms.

Also, setting i0 ≡ 1, the set I can be extended by deﬁning I0 ≡ {0} ∪ I. In this way, a supernumber can be written in the more compact form

X z = zαiα. (6.4) α∈I0

Furthermore, if z = P z i and w = P w i , their product can be written as α∈I0 α α β∈I0 β β

X zw = zαwβiαiβ. (6.5) α,β∈I0

Now, let α, β ∈ I and note that iαiβ = 0 when iα and iβ have a common factor iu, with u ∈ N. Moreover, when iαiβ does not vanish, it might still not be an element of the set

{iα : α ∈ I}, since permutations might be necessary to obtain an element of that set. However, because these permutations only introduce powers of negative one, there exists a

112 uniquely deﬁned γ ∈ I such that

σ(α,β) iαiβ = (−1) iγ, (6.6)

where σ(α, β) is the number of permutations necessary to “build” an element γ ∈ I from α and β. If such a relation holds, it is deﬁned that

α ∨ β ≡ γ (6.7)

σ(α,β) and, therefore, iαiβ = (−1) iα∨β.

To include the possibility of iαiβ = 0 in the deﬁnition of the operation ∨, it is deﬁned that

α ∨ β = ∅ if there exists no γ ∈ I0 such that Eq. (6.7) is satisﬁed. Then, the product of two supernumbers is written as

σ(α,β) X iαiβ = (−1) δα∨β,γiγ, (6.8) γ∈I0 where δα∨β,γ is the Kronecker delta.

It should be noted that Λ is a Z2-graded algebra. The elements that commute with each other are of the form X z = z0 + zαiα, (6.9) α∈I τ(α)≡0 mod 2

where τ(α) is the number of elements of α. Those supernumbers are called even supernumbers

and their set is denoted by Λeven. It is easy to verify that they commute with every element of Λ and that they form a commutative subalgebra.

113 On the other hand, the elements that anticommute with each other are of the type

X z = zαiα. (6.10) α∈I τ(α)≡1 mod 2

They are known as odd supernumbers and do not form a subalgebra. In fact, it is an immediate result that the product of two odd supernumbers is an even supernumber. The set of odd supernumbers is denoted by Λodd.

Observe that, if v ∈ Λodd ⊂ Λ,

1 X v2 = v v i i + i i = 0, (6.11) 2 α β α β β α α,β∈I τ(α),τ(β)≡1 mod 2 i.e., the square of odd supernumbers vanish.

Also, let N ∈ N and consider N + 1 elements zn ∈ ΛN such that znB = 0 for every n ∈ 1,...,N + 1. Then, it can be checked by direct computation that

N+1 Y zn = 0 (6.12) n=1 and, in particular, N+1 zS = 0 (6.13) for every z = zB + zS ∈ ΛN . This result can be also given for a z ∈ Λ such that zB = 0. In n(z) this case, there exists n = n(z) such that z ∈ Λn(z)−1 and, then, z = 0.

An important characteristic of the GA is that many of the properties of its elements depend solely on the properties of its complex body. For instance, a supernumber in Λ is invertible if and only if its body is diﬀerent from zero. In fact, if z = zB + zS ∈ Λ, it holds that

114 n(z) zS = 0. Then, if zB 6= 0, the inverse of z is given by

n(z) k X z z−1 = z−1 − S . (6.14) B z k=0 B

Conversely, assuming z is invertible, let its inverse be w = wB + wS ∈ Λ. Then,

zw = 1 ⇒ zBwB = 1 ⇒ zB 6= 0. (6.15)

It is common to endow Λ with an involution † for which the conjugation of a supernumber z given by Eq.(6.3) is † X π(α) z ≡ z0 + (−1) zαiα, (6.16) α∈I where π(α) = τ(α)(τ(α) − 1)/2. Observe that † can be characterized by the complex † † † † conjugation of the coeﬃcients zα, in = in, and (zw) = w z .

Of course, this choice for an involution is arbitrary. In Ref. [94], other examples of involutions in Λ are presented.

If z ∈ Λ is such that z† = z, it is said to be a real supernumber, or superreal; moreover, if z† = −z, it is said to be an imaginary supernumber [198]. Note that a real supernumber

generally does not belong to R. For instance, i1 + i2 is a real supernumber.

With an involution defined, it is also possible to introduce the idea of a non-negative and a non-positive supernumber. This is done by defining z ∈ Λ as a non-negative (resp. non- positive) supernumber if there exists w ∈ Λ such that z = ww† (resp. z = −ww†), and writing z 0 (resp. z 0). Observe that, with such a definition, only real supernumbers can be classified as non-negative or non-positive.

Furthermore, if one requires invertibility from z, then w is invertible and z is called a positive (resp. negative) supernumber, or superpositive (resp. supernegative), a property that is

115 denoted by z 0 (resp. z ≺ 0).

It is, then, clear that:

z 0 ⇔ zB > 0;

z ≺ 0 ⇔ zB < 0;

z 0 ⇒ zB ≥ 0;

z 0 ⇒ zB ≤ 0.

With that, it can be shown that every positive supernumber in Λ has a square root, meaning the analytic extension of the real square root to C\(−∞, 0]. To see that, let z = zB +zS ∈ Λ be a positive supernumber, which means that zB > 0. Thus,

zS zB 1 + (6.17) zB and " ∞ k+1# √ √ r z √ X 2 2k z z = z 1 + S = z 1 − − S . (6.18) B z B k + 1 k 4z B k=0 B Observe that the last sum converges because it has a ﬁnite number of non-zero elements.

Now, because the involution deﬁned in the GA induces a modulus that is given either by |z|2 = z†z or |z|2 = zz†, which, in general, is not a real number, the p-norm of a supernumber is introduced. For this purpose, let p ≥ 1 be a real number and, then, deﬁne the p-norm of a supernumber z ∈ Λ as  1/p X p kzkp =  |zα|  , (6.19) α∈I0 where | · | is the usual modulus of a complex number.

116 It can be checked that, if p = 1,

kzwk1 ≤ kzk1kwk1. (6.20)

Hence, Λ endowed with the 1-norm is a Banach algebra (BA). However, that is not the case if p > 1, and the p-norm of a product is not necessarily bounded by the p norm of each of its factors. In particular, when considering completions Λ(p) of Λ with respect to a p-norm, the resultant set Λ(p) might not be an algebra.

This chapter will deal with this problem when Λ(2) is introduced in Section 6.10. Before that, starting in the next section, the completion of Λ with respect to the 1-norm is studied.

6.2 1-Norm Completion of the Grassmann Algebra

From now until Section 6.9, it is studied the completion of Λ with respect to the 1-norm, which is the particular case of the p-norm of a supernumber z deﬁned in Eq. (6.19) with p = 1, i.e., X kzk1 = |zα|, (6.21) α∈I0 where | · | is the usual modulus of a complex number. When this completion, which is denoted by Λ(1), is endowed with the 1-norm, it becomes a complex vector space and has a BA structure.

It should be mentioned that some aspects of Λ(1) were already considered in the literature [199–204]. The present work, however, follows a diﬀerent approach.

Like in Λ, an element z ∈ Λ(1) is invertible if and only if zB 6= 0. To prove it, ﬁrst assume z ∈ Λ , which is a commutative BA. Then, let ϕ be a homomorphism between Λ (1)even (1)even and C.

117 It can be shown that ϕ(z) = zB. In fact, this follows from

2 2 ϕ(iα) = ϕ(iα) = ϕ(0) = 0 (6.22)

and ϕ(1) = 1. (6.23)

Therefore, by Gelfand0s theorem on invertibility in commutative BAs (see, e.g., Ref. [205]),

z is invertible if and only if ϕ(zB) 6= 0.

Now, in case z is not assumed to be an even supernumber, observe that, if z is invertible in

Λ(1), there exists w ∈ Λ(1) such that zw = wz = 1. In particular, zBwB = 1, which shows that zB 6= 0. Conversely, without loss of generality, let z = 1 + u + v ∈ Λ(1), where u + v is the soul of z, with u being its even part and v being the odd one. As already discussed, 1 + u is invertible. Then, h i z = (1 + u) 1 + (1 + u)−1 v (6.24)

is invertible if and only if q = 1 + (1 + u)−1 v is invertible. Note that q ∈ Λ . Thus, to S (1)odd complete the proof, it is left to show that an element z such that z ∈ Λ is invertible S (1)odd if z 6= 0. To do so, let z = 1 + v, where v ∈ Λ . Observing that the calculation shown B (1)odd in Eq. (6.11) is still valid in closures of Λ, it can be used that v2 = 0 and, then, it follows that w = 1 − v is the inverse of z. This ﬁnishes the proof that invertibility in Λ(1) can be reduced to invertibility of the body of the number.

With this result, it is possible to show that, while the soul of any supernumber in Λ is nilpotent, the soul zS of any element of Λ(1) is quasi-nilpotent. In fact, the spectral radius formula asserts that 1/n lim kznk = sup {|x| | x ∈ ρ(z )} , (6.25) n→∞ S 1 S

118 where ρ(zS) is the spectral radius of zS, which can be determined by ﬁnding the values of

λ ∈ C for which zS − λ is non-invertible. But, as just seen, a supernumber is non-invertible if and only if its body is non-invertible. Then,

ρ(zS) = 0, (6.26)

which implies that zS is quasi-nilpotent.

Now, let λ ∈ C be a complex variable and the complex power series

∞ X n f(λ) = cnλ , cn ∈ C, (6.27) n=0 be analytic in a neighborhood of the origin. Then, it follows directly from Eq. (6.26) that f(zS) converges in Λ(1).

More generally, if f(λ) is a complex analytic function for λ ∈ Ω, then f(z) converges in Λ(1)

for every z such that zB ∈ Ω. It can be seen by writing

∞ X f(n)(z ) f(z) = B zn. (6.28) n! S n=0

It should be noted that Eq. (6.28) often appears in the literature as a formal power series — see, e.g., Ref. [198]. Here, however, a meaning is given to it as an inﬁnite sum, in the sense

that it converges in Λ(1). Still on this discussion, a study of reproducing kernels associated with this type of extension in topological algebras was recently introduced [206].

Finally, observe that because of the results just presented, it is possible to conclude that

many analytic functions can be extended to Λ(1). For example, if the square root of a number is taken to be the analytic extension of the real square root to C \ (−∞, 0], then a positive supernumber z admits a square root.

119 6.3 Matrix Algebra and Extension of Toeplitz Matrices

In this section, Toeplitz matrices taking values in Λ(1) are studied. A Toeplitz matrix is a self-adjoint matrix with all elements of its main diagonal equal to each other. The aim here is to solve an extension problem for such matrices, namely how to obtain a Toeplitz matrix with a bigger size from a smaller one. The counterpart of this problem in the complex domain appears as an important player in many areas, like signal and image processing, system control, and in predictions of stationary processes of second order. Moreover, the center of a one-step extension in the classical theory is related to the concept of maximum entropy, and it is directly associated with the best estimation of parameters in stochastic processes, making it relevant in the solution to the Yule-Walker equations [207].

Before introducing Toeplitz matrices in the GA, some basic deﬁnitions and results in the

theory of matrices with entries in Λ(1) are studied. To begin, the conjugation † presented p×q in Section 6.1 can be extended to matrices M over the module Λ(1) . This is done in the following way ∗ † M = (mkj). (6.29)

In particular, (ML)∗ = L∗M∗ (6.30)

for matrices M and L of appropriate sizes.

p×q Also, the norm of a matrix M = (mjk) ∈ Λ(1) is deﬁned as

X kMk1 ≡ kmjkk1. (6.31) j,k

p×p Now, in the algebra of complex matrices, if a matrix M ∈ C is regular, meaning that all its main minor matrices are invertible, then M can be factorized as M = LDU, where D

120 is a diagonal matrix composed by invertible supernumbers, and L and U are, respectively, lower and upper-triangular matrices with main diagonals composed by ones. To see that a p×p similar result holds in this setting, let M = mjk ∈ Λ(1) be a matrix such that MB is

regular. Because of the regularity of MB, it can be assumed that (mkk)B > 0 for every

k ∈ {1, 2, . . . , p}, which implies that mkk is invertible. Thus,

  m B  11  M =   CE       (6.32) −1 1 0 m11 0 1 m B =      11  ,  −1   −1    Cm11 Ip−1 0 E − Cm11 B 0 Ip−1 where B, C, and E are block matrices. By successively repeating this process, it can, then, be concluded that, if M is a matrix such that MB is regular, M = LDU.

p×p Also, an invertible matrix M ∈ Λ(1) has an invertible body. The result that was just

proved, however, enables the proof of the converse, i.e., if MB is invertible, M is invertible. ∗ In fact, if MB is invertible, the body AB of A ≡ MMB is regular. Then, as it was just discussed, A = LDU. Because any lower or upper-triangular matrix with main diagonal composed by ones is invertible and D, which is composed of positive supernumbers, is also invertible, it can be concluded that A is invertible. Then, M admits an inverse given by −1 −∗ −1 p×p M = (MB) A . Therefore, any matrix M ∈ Λ(1) is invertible if and only if its body is invertible.

p×p Now, preparing for the extension of positivity to matrices of supernumbers, let M ∈ Λ(1) p×1 and c, d ∈ Λ(1) . Then, it can be shown that

d∗Mc = 0 (6.33)

121 for every c and d if and only if c∗Mc = 0 (6.34)

for every c. First, it is clear that, if the condition given by Eq. (6.33) holds, Eq. (6.34) also holds. To prove the converse, observe that, by the polarization identity,

3 1 X d∗Mc = c∗ + (−i)kd∗ M c + ikd . (6.35) 4 k=0

Because, now, Eq. (6.34) is assumed to hold, each term of the sum in Eq. (6.35) vanishes ∗ p×1 since they are of the type a Ma, where a ∈ Λ(1) .

Finally, it can be also shown that M = 0 whenever the condition given by either Eq. (6.33) or Eq. (6.34) holds. In fact, observe that, for the particular case where the j-th components of d and c are dj = δjr and cj = δjs, for some r, s ∈ {1, . . . , p}, Eq. (6.33) becomes just mrs = 0.

p×p With that, a matrix M ∈ Λ(1) is said to be a non-negative supermatrix (or simply super non-negative) if c∗Mc 0 (6.36)

p×1 for every c ∈ Λ(1) . Similarly, M is deﬁned to be superpositive if

c∗Mc 0 (6.37)

p×1 for every c ∈ Λ(1) such that cB is not the null element.

p×p With this deﬁnition, it follows that, if a supermatrix M ∈ Λ(1) is non-negative, then it is p×p self-adjoint and its body is non-negative. To see that, starting by assuming that M ∈ Λ(1) is non-negative. Thus, Eq. (6.36) holds, which implies that c∗Mc is a real supernumber for

122 p×1 every c ∈ Λ(1) , i.e., c∗Mc = c∗M∗c ⇒ c∗(M − M∗)c = 0. (6.38)

Then, M is self-adjoint, i.e., M∗ = M. Moreover, Eq. (6.34) implies that

∗ cBMBcB ≥ 0, (6.39)

p×1 for every c ∈ Λ(1) , which means that the body of M is non-negative.

Before introducing the problem of extension of Toeplitz matrices, one more result is pre- p×p sented. If M ∈ Λ(1) , the following statements are equivalent:

(i) M is superpositive;

(ii) M is self-adjoint and MB is positive;

(iii) M = LDU, where D is a superpositive diagonal matrix, L is a lower triangular matrix with main diagonal composed by ones, and U = L∗;

∗ ∗ p×p (iv) M = LL = UU , where L, U ∈ Λ(1) , and L and U are, respectively, lower and upper-triangular matrices and their main diagonals are composed by ones.

To prove this statement, start by observing that (ii) follows directly from the deﬁnition of superpositivity. Then, (i) implies (ii). Also, it follows from the discussion of Eq. (6.32) that (ii) implies (iii).

To show that (iii) implies (iv), observe that the matrix D in M = LDU is composed of positive supernumbers, which admit a positive square root. Then, D also has a superpositive diagonal square root. Denoting it by S, it holds that D = S2 and

M = LDU = LS2D = (LS)(LS)∗ = L0L0∗. (6.40)

123 In a similar way, one can start from a decomposition M = UDL, which is also valid, and conclude that M = U0U0∗.

p×1 Finally, the proof that (iv) implies (i) follows from the fact that, for every c ∈ Λ(1) such that cB is not the null element,

c∗Mc = c∗LL∗c = (L∗c)∗ (L∗c) 0 (6.41)

or, similarly, c∗UU∗c 0.

The extension of Toeplitz matrices can be now studied. The problem can be stated as: given a superpositive Toeplitz matrix

  r0 r1 . . . rN    †   r r0 . . . rN−1  T =  1  , (6.42) N  . . . .   . . .. .     † †  rN rN−1 . . . r0

† where r0, ··· , rN ∈ Λ(1) and r0 = r0, what are the conditions for a superpositive Toeplitz extension TN+1 be created from it? More speciﬁcally, what are the conditions that must be satisﬁed by a supernumber rN+1 ∈ Λ(1) to assure that

  TN bN+1 TN+1 =   , (6.43)  ∗  bN+1 r0 where bN+1 is a column matrix with coordinate bN+1 = (rN+1, rN , ··· , r1) ≡ (rN+1, bN ), is superpositive?

124 To start solving this problem, observe that TN+1 can be written as

      ∗ −1 ∗ −1 1 bN+1TN r0 − bN+1TN bN+1 0 1 0 TN+1 =       . (6.44)      −1  0 I 0 TN TN bN+1 I

∗ −1 Then, as already studied, TN+1 is a superpositive matrix if r0 − bN+1TN bN+1 is a positive supernumber. Moreover, writing

  r0 aN TN =    ∗  aN TN−1       (6.45) −1 −1 ∗ 1 aN T r0 − aN T a 0 1 0 =  N−1   N−1 N    ,      −1 ∗  0 I 0 TN−1 TN−1aN I it can be concluded that

∗ −1 † ∗ −1 r0 − bN+1TN bN+1 0 ⇔ (rN+1 − cN ) α (rN+1 − cN ) ≺ r0 − bN TN bN , (6.46)

−1 −1 ∗ −1 where α = r0 − aN TN−1aN and cN = aN TN−1bN . Furthermore, deﬁning real super- † ∗ −1 numbers ξ such ξ ξ ≡ r0 − bN TN bN , Eq. (6.46) can be rewritten as

† † (rN+1 − cN ) α (rN+1 − cN ) ≺ ξ ξ. (6.47)

This is, then, the solution to the problem of extension of Toeplitz matrices.

Obserce that 1/2 −1/2 α (rN+1 − cN ) ≺ ξ ⇒ rN+1 = cN + α ηξ, (6.48) where η is a supernumber such that η ≺ 1. Eq. (6.48) can be seen as a natural deﬁnition −1/2 of a disk with center in cN , left radius α , and right radius ξ. Such object is called a

125 superdisk.

Note that the “geometry” induced by the deﬁnition of positivity in the GA is diﬀerent from the one induced by the 1-norm. In fact, elements in the superdisk, in general, do not have a bounded norm. For instance, consider

1 a = (1 + λi ) λ ∈ . (6.49) 2 1 C

† Even though it is clearly inside the superdisk zz ≺ 1, its norm is kak1 = (1 + |λ|)/2, which can be arbitrarily large.

6.4 Realization Theory and Rational Functions

p×q p×q Denote by Γ (Ω) the space of power series whose coeﬃcients take value in Λ(1) that

converge on a neighborhood Ω of the origin in Λ(1). Then, the main goal of this section is to deﬁne and study realization theory and RFs of power series F ∈ Γp×q(Ω), i.e.,

∞ X n F (z) = z fn, (6.50) n=0

where the variable z varies in Ω. The approach followed here is similar to the one used in the quaternionic setting (see, e.g., [86, 208–210]) and, also, to the Fueter variables on BAs presented in Section 5.7. An important distinction is the fact that, here, the focus is on functions that are also invertible at the origin.

To start, the Cauchy-Kovalevskaya (CK) product is introduced in this setting. For that, let F and G be two power series of the type given by Eq. (6.50). Then, the CK product is deﬁned as   X n X F?G(z) ≡ z  fugn−u . (6.51) n∈Z u∈Z

126 The product is denoted by ?, and not by , so it can be diﬀerentiate from the CK product deﬁned in the previous chapter.

Note that such a product reduces to the pointwise product in the case of z = λ ∈ C. Moreover, for F (z) invertible, the star and the regular products are related by

X n F?G(z) = z F (z)gn n∈Z X −1 n = F (z) F (z) z F (z)gn (6.52) n∈Z = F (z) G F (z)−1zF (z) .

Now, F ∈ Γp×q(Ω) is said to admit a realization if it can be represented in the form

−? F (z) = D + zC ? (IN − zA) B, (6.53)

−? where D ≡ F (0), A, B, C are supermatrices of appropriate sizes, and (IN − zA) ≡ P∞ n n n=0 z A .

If the matrix D in Eq. (6.53) is invertible, it can be veriﬁed by direct computation that

F (z)−? = D−1 − zD−1C? (I − zA×)−?BD−1 (6.54) is a realization of F −?, with A× = A − BD−1C. (6.55)

Moreover, it can checked that, if

−? Fj(z) = Dj + zCj ? (INj − zAj) Bj, (6.56)

127 j = 1, 2, correspond to two realizations of compatible sizes, a realization of F1(z) ?F2(z) is given by     A B C B D  1 1 2  1 2 A =   ,B =   , 0 A B 2 2 (6.57) C C1 D1C2 ,D = D1D2

and a realization of F1 + F2 is given by

    A 0 B  1   1 A =   ,B =   , 0 A B 2 2 (6.58) C = C1 C2 ,D = D1 + D2.

Now, deﬁning a k-th order polynomial in Γp×q(Ω) as a ﬁnite sum of the form

k M(z) = M0 + zM1 + ··· + z Mk, (6.59)

p×q where M0,...,Mk ∈ Λ(1) , it is clear that any polynomial in z admits a realization.

It is also of interest to deﬁne a pair (C,A) as an observable pair of matrices if

∞ u ∩u=0 ker CA = {0} . (6.60)

Moreover, a pair (A, B) is said to be controllable if

ξ(I − λA)−1B ≡ 0 ⇒ ξ = 0, (6.61)

where λ ∈ R. Furthermore, a realization is called minimal if the pair (C,A) is observable and the pair (A, B) is controllable.

128 With that, it can be shown that two minimal realizations

−? F (z) = Dj + zCj ? (INj − zAj) Bj, j = 1, 2, (6.62)

of a function F are similar. To do so, ﬁrst observe that D1 = D2. Then, it is possible to write F (x) − F (y) −1 −1 = C1(IN1 − xA1) (IN1 − yA1) B1 x − y (6.63) −1 −1 = C2(IN2 − xA2) (IN2 − yA2) B2,

0 where x, y ∈ R and the understanding that the left-hand side is equal to F (x) if x = y. Moreover, deﬁne the operators

−1 −1 U (IN1 − yA1) B1ξ ≡ (IN2 − yA2) B2ξ, (6.64)

and −1 −1 V (IN2 − yA2) B2ξ ≡ (IN1 − yA1) B1ξ, (6.65)

q where ξ ∈ Λ(1). The fact that the pairs (Cj,Aj) are observable assures that such operators are well deﬁned. From the above deﬁnitions, and because the pairs (Aj,Bj) are controllable, it holds that

UV = IN2, (6.66) and

VU = IN1, (6.67)

where IN denotes the N × N identity operator.

N Now, deﬁning elements ej, j = 1,...,N, of modules Λ(1) with components given by

ej k = δjk, (6.68)

129 it can be written that N1 −1 X (IN1 − yA1) B1ξ = ejαj, (6.69) j=1 where αj ∈ Λ(1), and

    N1 N1 N1 N2 X X X X U  ejαj = U ej αj =  ukjej αj. (6.70) j=1 j=1 j=1 k=1

˜ Hence, U can be represented by a N2×N1 matrix U = ukj . Similarly, V can be represented by a N1 × N2 matrix V˜ . To conclude the proof, because (A1,B1) is controllable, N2 ≤ N1.

However, because (A2,B2) is also controllable, N1 ≤ N2. Therefore, N1 = N2 and two minimal representations are similar.

Presented some deﬁnitions and results on realizations, RFs can be discussed. As mentioned in Section 5.7, a RF F ∈ Γp×q(Ω) is a quotient of polynomials. In particular, a RF F ∈ Γp×r can be written as a ﬁnite product of the type

−? F (z) = M1(z) ?M2(z) ?M3(z), (6.71)

p×q q×q q×r where M1 ∈ Γ , M2 ∈ Γ , and M3 ∈ Γ are polynomials. Then, it is clear that every realization is a RF. Therefore, because every polynomial admits a realization and the inverse of a realization also admits a realization, the concepts of RFs and realizations are equivalent.

Finally, it can be checked by direct computation that, because every RF admits a realization, and vice versa, they are characterized by the Taylor expansion

∞ X F (z) = D + znCAn−1B, (6.72) n=1 where the matrices A, B, C, and D are the ones deﬁned in Eq. (6.53).

130 6.5 Rational Schur-Grassmann functions

The purpose of this section is to start the study of Schur analysis and related topics in

Λ(1), a subject that will continue to be explored until Section 6.9. To put the study into perspective, a brief review of it in the complex setting is presented.

Schur analysis is a part of function theory in the open unit disk D or a half-plane. It is a rich and vastly developed ﬁeld with numerous applications, which include, but are not limited to, signal processing [211], fast algorithms [212], and linear systems [213]. It originated with Schur in 1979 [214, 215], although this area can be traced back to Stieltjes’s work in 1894 [216]. A collection of original papers on the topic can be found in Ref. [217]. The Hardy space and Blaschke factors are important players in this domain [218, 219], as well as the Wiener algebra and RFs.

The Hardy space of the unit disk H2 is the Hilbert space (HS) of power series f(λ) = P∞ a λn, λ ∈ , such that kfk2 ≡ P∞ |a |2 < ∞. From the signal processing n=0 n C H2 n=0 n point of view, it can be interpreted as the space of frequencies (z-transforms) of finite energy discrete signals. Such an interpretation motivates various interpolation problems in the Hardy space and related spaces. The Nevanlinna-Pick and the Carathéodory-Fejér problems

are two examples of it. The former consists on, given λ1, . . . , λN in D and complex numbers

ω1, . . . , ωN , describing the set of all Hardy functions f such that f(λj) = ωj for j = 1,...N. The latter, on the other hand, refers to the problem of fixing the first N derivatives of a function at a given point. Moreover, for λ = 0, the Carathéodory-Fejérproblem becomes trivial for Hardy functions — since the coefficients an of f are known. Nevertheless, this is a problem of central importance in the class of functions with a positive real part in the open unit disk, which is related to the theory of extension of Toeplitz matrices and has applications on the prediction theory of second-order stationary processes.

For both the Nevanlinna-Pick and the Carath´eodory-Fej´erproblems, however, additional

131 metric constraints are imposed to f, such as requiring that it should take contractive values in the open unit disk. Functions satisfying such conditions are called Schur functions, and they are the transfer functions of dissipative systems [187, 220–224].

Schur functions can be characterized in a number of equivalent ways, including the just mentioned contractivity in the Hardy space. In the rational setting, which is our starting point here, a Schur function S can be deﬁned as a matrix-valued RF which is analytic at inﬁnity and has minimal realization given by

S(λ) = D + C(λI − A)−1B (6.73) such that  ∗       AB H 0 AB H 0               ≤   (6.74) CD 0 I CD 0 I for some (not uniquely deﬁned) H < 0. This characterization results from the positive real lemma, also known as Kalman-Yakubovich-Popov theorem [225–227], and its generalization [228]. For the description of all H satisfying Eq. (6.74), see Ref. [226, 227].

It should be noticed that Eq. (6.73) presents a diﬀerent expression for a realization than the one studied in the previous section, which can be written as

S(λ) = H + λG(I − λT )−1F, (6.75) which, in general, is not analytic at inﬁnity. However, both expressions are equivalent if the matrix A is invertible. In fact, let H = D − CA−1B, G = −CA−1, T = A, and F = AB.

132 Then, Eq. (6.75) can be rewritten as

S(λ) = D − CA−1B − λCA−1(I − λA)−1AB

= D − CA−1B + CA−1(λA − I)−1(I + λA − I)B (6.76) = D + CA−1(λA − I)−1B

= D + C(λI − A)−1B.

With that, Eq. (6.73) with matrices conditioned to Eq. (6.74) is taken as the primary p×q deﬁnition of a rational Schur-Grassmann (SG) function. To be precise, a Λ(1) -valued RF with realization given by −? S(z) = D + C? (zIN − A) B (6.77)

N×N will be called a SG function if there exists a Hermitian strictly negative matrix H ∈ Λ(1) , i.e., H ≺ 0, such that

 ∗       AB H 0 AB H 0                 . (6.78) CD 0 I CD 0 I

p×q Now, let S be a Λ(1) -valued RF of the type given by Eq. (6.77). Then, it can be veriﬁed that S is a SG function if and only if its body part SB is a Schur function. This allows the direct translation to SG functions of a number of properties of Schur functions. Two examples of that can be given:

Complex rational Schur functions S(λ) can be characterized by S(λ)S(λ)∗ ≤ I for |λ| ≤ 1. Then, it can be checked that rational SG functions S(z) are such that S(z)S(z)∗ I for zz† 1.

133 Complex rational Schur functions S(λ) can characterized by the fact that the kernel

∞ X λn (I − S(λ)S(ω)∗) ωn (6.79) n=0

is positive deﬁnite. Also, rational SG functions S(z) are such that the kernel

∞ X zn (I − S(z)S(w)∗)(w†)n (6.80) n=0

is superpositive.

However, not every result from complex analysis holds in the present setting by a simple “reduction to the body” study of SG functions. As it will be seen in more details later, interpolation problems are an example of that.

Now, consider the Hermitian form

∞ X ∗ [F,G] ≡ gnfn, (6.81) n=0 where ∞ X n F = z fn (6.82) n=1 and ∞ X n G = z gn (6.83) n=1 q×q are such that [F,F ] and [G, G] converge in Λ(1) . Note that the restriction of Eq. (6.81) to its body corresponds to the matrix-valued Hermitian form associated to the Hardy space of p×q C -valued functions analytic in the open unit disk.

With this deﬁnition, another result that follows from the analysis of complex Schur functions

134 states that a RF S is a SG function if and only if

[MSF,MSF ] [F,F ] , (6.84)

where MS is the CK multiplication operator, deﬁned by MSF ≡ S?F .

p×q Finally, let S be a Λ(1) -valued RF given by

S(z) = s0 + zs1 + ··· , (6.85)

p×q where s0, s1,... ∈ Λ(1) . Moreover, let LN denote the lower triangular matrix

  s0 0 ··· 0 0      s1 s0 ··· 0 0  L =   . (6.86) N  . . . . .   ......      sN sN−1 ··· s1 s0

∗ Then, it follows from complex analysis that S is a SG function if and only if LN LN I for every N ∈ N.

6.6 Wiener-Grassmann algebra

In this section, it is introduced the Wiener algebra associated with the GA Λ(1), or simply the Wiener-Grassmann (WG) algebra. In the complex case, the characterization of boundary values of Hardy functions can be a quite challenging problem. Then, it is often desirable to p consider functions in W+, which is a subset of the Wiener algebra

 

p  it X int p×p X  W = f(e ) = e fn t ∈ R, fn ∈ C , kfnk < ∞ , (6.87)

 n∈Z n∈Z 

135 where k · k is a complex matrix norm. Wp is endowed with the pointwise multiplication of functions.

Some facts about Wp are relevant for the present discussion. For instance, a function f ∈ Wp is said to be strictly positive if f(eit) > 0 for every real t. Moreover, the Wiener-L´evytheorem assures that f has an inverse in Wp if and only if f(eit) 6= 0 for every real t, i.e., invertibility in the algebra is equivalent to pointwise invertibility.

p p Two important subalgebras of W are the aforementioned W+, composed by functions f of the type ∞ it X int f(e ) = e fn, (6.88) n=0 p and W−, which contains functions of the type

0 it X int f(e ) = e fn. (6.89) n=−∞

p p p p There are elements in W+ (resp. W−) that have inverse in W+ (resp. W−). They are p denoted by f+ (resp. f−). Furthermore, some functions in W can be factorized as f = p f+f−. An important theorem states that f ∈ W is strictly positive if and only if it has ∗ ∗ such a factorization and it is characterized by f− = f+, where f+ denotes the adjoint of f+.

Now, some of these results and deﬁnitions are introduced in the present setting. To start, the WG algebra is deﬁned as

 

p  X n p×p  W = f(z) = z fn z ∈ Λ , fn ∈ Λ , k [f, f] k1 < ∞ , (6.90) G (1) (1)  n∈Z  where, the Hermitian form [·, ·] is deﬁned as

X ∗ [f, g] = gnfn. (6.91) n∈Z

136 This algebra is endowed with the CK product introduced in Eq. (6.51), i.e.,

  X n X f ? g(z) ≡ z  fugn−u , (6.92) n∈Z u∈Z

p for any two functions f and g in WG.

p p An import subalgebra of WG for this work is the set WBP , which will be called the Wiener- Bochner-Phillips algebra and is deﬁned by

 

p  X int p×p X  W = f(t) = e fn t ∈ R, fn ∈ Λ , kfnk1 < ∞ , (6.93) BP (1)  n∈Z n∈Z 

where i ∈ C is the imaginary unit. Note that the CK product is reduced to the pointwise p product in WBP .

p p With this deﬁnition, it can be shown that a function f ∈ WBP has an inverse in WBP if

and only if f(t) has an inverse in Λ(1), for every t. First, it is clear that f(t) is invertible for p every t if f is invertible in WBP . Moreover, the converse is just an adapted version of the original result presented by Bochner and Phillips in Ref. [229, Theorem 1] to the case where p×p the coeﬃcients are elements of Λ(1) instead of a generic noncommutative ring.

p it p Now, let f ∈ WG and fBP ≡ f(e ) ∈ WBP , where t ∈ R. Then, the following statements are equivalent:

p (i) f is invertible in WG;

p (ii) fBP is invertible in WBP ;

p (iii) the body of fBP is invertible in the classical Wiener algebra W ;

(iv)( fBP )B (t) 6= 0 for every t.

137 To see that (i) is equivalent to (ii), observe that, if (i) holds, there exists a function g ∈ p it WG such that f ? g(z) = 1 for every z. In particular, this holds for points z = e , i.e., it fBP gBP (t) = 1 for every t, where gBP ≡ g(e ). Conversely, if (ii) is true, there exists

X int p gBP = e gn ∈ WBP (6.94) n∈Z such that fBP gBP = 1. Then, it holds that

 X  0, n 6= 0 fugn−u = . (6.95) u∈Z  1, n = 0

Finally, consider the “extension” of gBP to WG

X n g = z gn, (6.96) n∈Z where z ∈ Λ(1) such that zB 6= 0. It follows that f ? g = 1.

Moreover, to prove that (ii) is equivalent to (iii), observe that fBP gBP = 1 ⇒ (fBP )B (gBP )B = p 1 and (fBP )B , (gBP )B ∈ W . For the converse, assume that (iii) holds, which implies that p fBP has an inverse in Λ(1). In this case, as already discussed, fBP is invertible in WBP .

Finally, (iii) being equivalent to (iv) is just a restatement of the classical Wiener-L´evytheorem.

Before concluding this section, in a similar way done in the complex setting, the subsets

p p WG+ = f ∈ WG fn = 0, n < 0 (6.97) and p p WG− = f ∈ WG fn = 0, n > 0 (6.98)

138 are introduced.

p With that, a weak condition for the invertibility of a function in WG+ is presented: a p p function f ∈ WG+ has an inverse in WG+ if and only if [f(z)]B 6= 0 for every z such that kzk1 ≤ 1.

p p To show that, assume that f is invertible in WG+. Then, there exists g ∈ WG+ such that

∞ ! ∞ ! X m X n z fm ? z gn = 1 (6.99) m=0 n=0 for every z ∈ Λ(1). In particular, restricting this equation to its body,

∞ ! ∞ ! X m X n zB (fm)B zB (gn)B = 1, (6.100) m=0 n=0

p i.e., [f(z)]B 6= 0. Conversely, let f ∈ WG+ and [f(z)]B 6= 0 for every z such that kzk1 ≤ 1. p In particular, for z = ξ ∈ C with |ξ| ≤ 1, fB(ξ) is an invertible element of W+. Hence, as p p already proved, f is invertible in WG. Therefore, if g ∈ WG is its inverse,

∞ !   X m X n z fm ?  z gn = 1. (6.101) m=0 n∈Z

p Furthermore, using, again, the fact that fB(ξ) is invertible in W+, it can be concluded that

(gn)B = 0 for every n < 0.

It should be noticed that a similar result to the one just presented holds for the invertibility in p WG−. Moreover, a strong version of it, if it exists in the present setting, should characterize p p functions in WG+ that have inverse in WG+, as it is done in complex analysis.

139 6.7 Reproducing Kernel Banach Modules and Interpolation

A module is said to admit a reproducing kernel if there exists a positive deﬁnite function K(z, w) and a Hermitian form [·, ·] such that every function in such a set can be pointwise evaluated as f(z) = [f(·),K(·, z)] . (6.102)

P∞ n In the complex case, this Hermitian form on a space of power series f = n=0 z an, with P∞ 2 an ∈ C and n=0 |an| < ∞, is a map into C and coincides with the usual inner product

∞ X [f, g] = bnan, (6.103) n=0

P∞ n where g = n=0 z bn, with bn ∈ C. Note that, with respect to this product, such a space of power series is a complex BA. Moreover, a similar deﬁnition could be given for matrix-valued coeﬃcient an of f.

Now, when replacing the complex coeﬃcients in Eq. (6.102) by elements of Λ(1), it becomes p clear that one type of power series of interest is the one deﬁned by the Wiener algebra WG+ studied in the previous section.

Since a Hermitian form of the type given by Eq. (6.103) is necessary, besides the CK product, this set is also endowed with the form deﬁned in Eq. (6.81). Observe that, if

∞ X K(z, w) = zn(w†)n, (6.104) n=0

the point evaluation of a function f ∈ WG+ is

f(z) = [f(·),K(·, z)] . (6.105)

140 p Moreover, the notion of orthogonality in WG+ is deﬁned according to the Hermitian form p p in Eq. (6.81), i.e., f ∈ WG+ is said to be perpendicular to g ∈ WG+ if [f, g] = 0.

Solutions to interpolation problems are the main concern of this section. For that, the p introduction of the notion of linear independence is necessary. Since WG+ is a module (and not a linear space) such a notion is, in general, delicate. However, consider the set G = {f , . . . , f } ⊂ Wp such that (f ) ,..., (f ) are linearly independent. Then, it is 1 N G+ 1 B N B possible to show that G is a basis for span(G). Indeed, it is only needed to prove that, given

coeﬃcients cj ∈ Λ(1), if

f1c1 + ... + fN cN = 0, (6.106)

then c1 = c2 = ... = cN = 0. For this purpose, assume that f1, . . . , fN are such that their

body are linearly independent, and Eq. (6.106) is satisﬁed by coeﬃcients cj 6= 0. Then,

observe that the restriction of Eq. (6.106) to its body implies that the body of every cj must be null. Moreover, writing X cj = cjαiα, (6.107) α∈I

let mj be the minimum value of τ(α) for which cjα 6= 0. Also, let m = min{m1, m2, . . . , mN }. As a result, for every α such that τ(α) = m, Eq. (6.106) implies that

f1c1α + ... + fN cN α = 0. (6.108)

Since at least one cjα is not null, Eq. (6.108) contradicts the fact that f1, . . . , fN are linearly

independent. Therefore, Eq. (6.106) can only be satisﬁed if c1 = c2 = ... = cN = 0.

Now, it is introduced the function

−? −1 −∗ ∗ Θ(z) ≡ Ip − (1 − z)C? (Iq − zA) P (Iq − A) C J, (6.109)

141 p×q q×q p×p where (C,A) ∈ Λ(1) × Λ(1) is an observable pair, J ∈ Λ(1) is a signature matrix, and q×q P ∈ Λ(1) is an invertible self-adjoint matrix. This function is a fundamental object for the results presented here.

Furthermore, the CK product can be also deﬁned for power series with the variable z placed on the right-hand side of the coeﬃcients as

∞ ! ∞ ! ∞ n ! X n X n X X n fnz ?r gnz ≡ fugn−u z . (6.110) n=0 n=0 n=0 u=0

As a result, the expression

∞ n X n ∗ † −? −1 ∗−?r ∗ z (J − Θ(z)JΘ(w) ) w = C? (Iq − zA) P (Iq − wA) ?r C (6.111) n=0

holds if and only if the Stein equation

P − A∗PA = C∗JC (6.112)

−1 −1 holds. To verify it, consider z = λ and w = ω in C. Writing α(λ) = C(Im −λA) P (Im − −∗ ∗ A) and β = (Iq − A) P (Iq − A), observe that

∗ h −1 −1 −∗ ∗ i J − Θ(λ)JΘ(ω) = J − Ip − (1 − λ)C(Iq − λA) P (Iq − A) C J J×

h −1 −1 ∗−1 ∗i × Ip − (1 − ω)JC(Iq − A) P (Iq − ωA) C (6.113)

= α(λ)[β − λβω + (1 − λ)(1 − ω)(P − A∗PA − C∗JC)] α(ω)∗,

which proves the result for z = λ. For an arbitrary z ∈ Λ(1), the above calculation follows in a similar way. Because of this result and for reasons that will become clear later, the Stein equation is assumed to hold hereby.

The goal of the remaining of this section is the solution to the following interpolation problem:

142 p×q q×q ∗ ∗ Given an observable pair (C,A) ∈ Λ(1) × Λ(1) such that C C = P − A PA, ﬁnd every f that satisﬁes (C∗ ?F )(A∗) = X. (6.114)

First, it can be checked by direct computation that the function

∞ −? −1 X n n −1 Fmin = C? (Iq − zA) P X = z CA P X (6.115) n=0

is a particular solution to Eq. (6.114). As a result, if F is another solution, then G = F −Fmin satisﬁes the homogeneous problem, i.e.,

(C∗ ?G)(A∗) = 0. (6.116)

P∞ n Now, writing G = n=0 z gn, Eq. (6.116) is equivalent to

∞ X ∗ n ∗ (A ) C gn = 0, (6.117) n=0

which, in turn, is equivalent to

G, C ? (I − zA)−?ξ = 0, (6.118)

p −? for every ξ in WG+. In other words, G is orthogonal to C? (I − zA) ξ.

Continuing, let H(Θ) be the set of functions F of the type

F = C? (I − zA)−?ξ. (6.119)

143 Such a space is associated with the reproducing kernel

−? −1 ∗−?r ∗ KH(Θ)(z, w) = C? (Iq − zA) P (Iq − wA) ?r C , (6.120) with P − A∗PA = C∗C. Then, using Eq. (6.111),

∞ X n ∗ † n KH(Θ)(z, w) = z [I − Θ(z)Θ(w) ](w ) ξ. (6.121) n=0

p Finally, observe that WG+ can be decomposed as the direct sum

p p WG+ = ΘWG+ ⊕ H(Θ). (6.122)

In fact, using Eq. (6.121), the kernel K deﬁned in Eq. (6.104) satisﬁes

∞ ∞ X n † n X n ∗ † n K(z, w)ξ = z I(w ) ξ = z [Θ(z)Θ(w) ](w ) ξ + KH(Θ)(z, w)ξ (6.123) n=0 n=0

p×1 for every ξ ∈ Λ(1) . The two terms on the right-hand side are orthogonal to each other. The P∞ n ∗ † n p term n=0 z [Θ(z)Θ(w) ](w ) ξ belongs to ΘWG+ and the term KH(Θ)(z, w)ξ belongs to H(Θ). By Eq. (6.118), the two sets are orthogonal to each other. Therefore, Eq. (6.122) holds.

As a result, all solutions to Eq. (6.114) can be written as

F = Fmin + Θ ?H, (6.124)

where H is an arbitrary element of WG+.

144 6.8 Nevanlinna-Pick Interpolation

In Section 6.5, the original version of the Nevanlinna-Pick (NP) interpolation was brieﬂy

introduced in the complex setting. Now, a version of it in Λ(1) is studied. Given N points zk

in the open unit superdisk and N values sk in Λ(1), the NP problem consists of characterizing all SG functions S satisfying

S(zk) = sk (6.125) and such that the Pick matrix P is superpositive, i.e.,

P = pjk ≡ pk(zj; sj) 0, (6.126)

where pk(z; s) is deﬁned as

† † −? pk(z; s) ≡ (1 − ssk) ? (1 − zzk) (6.127) for every k ∈ {1,...,N}.

Following the solution to the complex case [187, 230], consider the function Θ deﬁned in Eq. (6.109) with

 †  z1       1 ··· 1 1 0  ..      A ≡  .  ,C ≡   , and J ≡   . (6.128)   s† ··· s† 0 −1  †  1 N zN

Note that P − A∗PA = C∗JC.

145

As a result, 1 −s ? Θ(z) equals k z=zk

−? −1 −∗ ∗ 1 −s − (1 − z) 1 −s C? (I − zA) P (I − A) C J . (6.129) k k N N z=zk

Moreover, because

C = † † , (6.130) 1 −sk 1 − sks1 ··· 1 − sksN it holds that −? [k] 1 −s C? (I − zA) = P , (6.131) k N z=zk

[k] [k] where P denotes the k-th row of P . Hence, if IN denotes the k-th row of IN ,

[k] −∗ ∗ 1 −s ? Θ(z) = 1 −s − (1 − z )I (I − A) C J k k k N N z=zk −1 [k] ∗ (6.132) = 1 −sk − (1 − zk)(1 − zk) IN C J

= 0

[k] ∗ since IN C J = 1 −sk . Therefore,

1 −s ? Θ(z) = 0 (6.133) k z=zk for every k ∈ {1,...,N}.

Now, let σ be any SG function and consider the term

  σ(z)   1 −S(z) ? Θ(z) ?   . (6.134) 1

146 Note that such a product is null at z = zk. Also, writing

  a(z) b(z)   Θ(z) =   , (6.135) c(z) d(z) it holds that

−? s = (a ? σ(z) + b(z)) ? (c ? σ(z) + d(z)) , (6.136) k z=zk provided that the factor c ? σ(zk) + d(zk) is CK invertible. In fact, that is the case since only the body of the factor needs to be invertible, and a theorem from complex analysis assures that it is. Observe that Eq. (6.136) allows to write the function S as

S ≡ TΘ(σ). (6.137)

Then, for S to be a solution to the NP problem, it just needs to be shown that, if σ in Eq. (6.137) is a SG function, S is also a SG function. But this is, in fact, the case because, since

  σ ∗ † ∗   † Θ JΘ J ⇒ σ 1 ? Θ JΘ ?   σ σ − 1, (6.138) 1 it holds that

  a ? σ + b † †   (a ? σ + b) −(c ? σ + d)   0 c ? σ + d

(a ? σ + b)†(a ? σ + b) (c ? σ + d)†(c ? σ + d) (6.139) † (a ? σ + b) ? (c ? σ + d)−? (a ? σ + b) ? (c ? σ + d)−? 1

S†S 1, which characterizes a SG function.

147 Finally, to conclude the solution, it just needs to be shown that all solutions are of the type given by Eq. (6.137), i.e., if a SG function S is a solution to the NP problem, then there exists a SG function σ such that S is given by Eq. (6.137). To see that this is the case, observe that there always exists a function σ given by

−1 σ = TΘ (S), (6.140)

even if it, in principle, does not belong to the set of SG functions. However, because the restriction to the body corresponds to the NP interpolation problem in the complex setting

(with Pick matrix PB > 0), it can be stated that σB is a Schur function. Therefore, σ is a

SG function, and all solutions to the NP problem in Λ(1) are given by Eq. (6.137).

6.9 Schur Algorithm

In Ref. [214], using Schwarz’ lemma, Schur associated to a Schur function s(λ) a sequence,

ﬁnite or inﬁnite, of Schur functions s0, s1 ... with the recursion

s0(λ) = s(λ) (6.141) sn(λ) − sn(0) sn+1(λ) = , λ(1 − sn(λ)sn(0))

for n = 0, 1,.... Such a recursion ends at a rank N if |sN (0)| = 1, and this happens if and

only if s is a ﬁnite Blaschke product. The numbers ρn = sn(0), n = 0, 1,... are called the Schur coeﬃcients of s. They lead to a continued fraction expansion of s, and prove more appropriate than the Taylor series of s to solve various approximation problems [187, 212].

Now, for ρ ∈ , let D     1 1 ρ λ 0 Θρ(λ) ≡     . (6.142) p 2     1 − |ρ| ρ 1 0 1

148 Then, expressing sn in terms of sn+1, Eq. (6.141) can be rewritten as

s (λ) = T (s (λ)), (6.143) n Θρn(λ) n+1

with ρn = sn(0). The matrix-function Θρn(λ) is, then, J-inner in the open unit disk. By

the latter, it is meant that Θρn(λ) is J-contractive in the open unit disk, i.e.,

∗ Θρn(λ)JΘρn(1/λ) ≤ J (6.144)

for λ ∈ D, and Θρn(λ) is J-unitary (or symplectic) at the boundary of the unit disk, i.e.,

∗ Θρn(λ)JΘρn(1/λ) = J (6.145)

for λ ∈ ∂D.

Observe that Θρn(λ) remains J-inner if it is multiplied by a J-unitary constant, say Xn, on the right. Thus, Eq. (6.143) can be rewritten as

sn(λ) = T (T −1(sn+1(λ))). (6.146) Θρn(λ)Xn Xn

Since Xn is J-unitary, the function T −1(sn+1(λ)) is still a Schur function. This fact was Xn used in Ref. [231, §3] to develop the Schur algorithm in the matrix-valued case — see, in particular, Eq. (4.13) in that paper.

A particular choice of Xn leads to

  1     1 −ρn ρn Mn ≡ Θρn(λ)Xn = I2 − (1 − λ) 2 , (6.147) 1 − |ρn|

149 which allows the Schur algorithm to be rewritten as

σ0(λ) = s(λ) (6.148) σ (λ) = T −1(σ (λ)) n+1 Mn(λ) n

for n = 0, 1,... This recursion with the counterpart of Eq. (6.147) is how the Schur algorithm

is deﬁned in Λ(1). More precisely, using Eq. (6.109) with

    1 1 0 A = 0,C =   and J =   , (6.149)  †    ρn 0 −1 the Stein equation becomes † P = 1 − ρnρn. (6.150)

Moreover, if  † −1  a(z) ≡ 1 − (1 − z)(1 − ρnρn) ,   † −1  b(z) ≡ (1 − z)(1 − ρnρn) ρn, (6.151)  c(z) ≡ −(1 − z)ρ† (1 − ρ ρ† )−1,  n n n   † † −1  d(z) ≡ 1 + (1 − z)ρn(1 − ρnρn) ρn,

a version of Mn in Λ(1) can be written as

    1 † −1 a(z) b(z) Mn(z) = I − (1 − z)   (1 − ρnρn) 1 −ρ =   . (6.152)  †  n   ρn c(z) d(z)

Finally, the Schur algorithm for SG functions can be stated: If S is a SG function. Then, the recursion

σ0(z) = S(z) (6.153) σ (z) = T −1(σ (z)), with ρ = σ (0) n+1 Mn(z) n n n

150 † deﬁnes a family of SG functions, provided that P = 1 − ρnρn 0.

6.10 Fock-Bargmann-Segal Space

The complex Fock-Bargmann-Segal (FBS) space associated with `2, i.e., the HS with reproducing kernel [232] α α hz,wi X z w e `2 = (6.154) α! α∈` corresponds to function theory in a (countably) inﬁnite number of commuting complex variables. In Eq. (6.154), the several complex variables notation was used and, then,

z = (z1, z2,...) ∈ `2(N), where ` is the family of sequences

α = (α1, α2,...), αj ∈ N0 (6.155)

α α1 α2 for which at most a ﬁnite number of αj’s are diﬀerent from 0. Also, z = z1 z2 ... and

α! = α1!α2! ....

In the noncommutative setting, the commuting variables are replace by noncommuting ones

and, then, a different set of indexes è is needed. An element α ∈ è is given by

α = ((α1, n1), (α2, n2),..., (αm, nm)), (6.156)

where αu ∈ N for every u ∈ {1, . . . , m} and nv 6= nv+1 for every v ∈ {1, . . . , m − 1}. Moreover, the new kernel is written as

X zαwα, (6.157) α∈`e

α α1 α2 α α2 α1 where z = zn1 zn2 ... and w = ... wn2 wn1 .

151 Note that the left concatenation gives èa monoid structure. Furthermore, it defines a partial order as follows: for α, β ∈ è, it is said that β ≤ α if there is γ ∈ è such that α = βγ. There

exists, then, a parallel between the indexes in `e and the ones in I0. Motivated by it and envisioning the construction of the counterpart of the FBS space in the GA, the inner product

X hz, wi = zαwα (6.158) α∈I0

is deﬁned for elements of Λ. Observe that

hw, zi = hz†, w†i. (6.159)

Also, the inner product deﬁned in Eq. (6.158) induces the 2-norm. Then, it can be extended

from Λ to Λ(2). Moreover, Λ(2) endowed with the inner product deﬁned in Eq. (6.158)

and the 2-norm is a HS. For simplicity of notation, such a HS is simply denoted by Λ(2). By analogy with the noncommutative setting, as already discussed, and also noting the deﬁnition in Ref. [233], Λ(2) is called the FBS space.

This section, then, starts a study, which will last until Section 6.12, of Λ(2). This study

considers functions f : I → Λ(2), where I is the domain of f, usually C or R. Whenever it

is written that f ∈ Λ(2), it is meant that f(x) ∈ Λ(2) for every x ∈ I.

As in the examples of FBS spaces in diﬀerent settings, it is possible to deﬁne the left multipli-

cation operation Mf in Λ(2). However, the multiplication might not be a law of composition

in Λ(2). If that is the case, Mf is unbounded for an arbitrary f ∈ Λ(2). One can, then, use two diﬀerent approaches. In one of them, which will be considered in the next section, such

operators can be studied in the space of stochastic distributions S−1, where the product

between two functions deﬁnes a new element of the space. In the other approach, Mf is

restricted to functions f ∈ Λ(1) ⊂ Λ(2) Eq. (6.191) assures that Mf is bounded in this case,

152 i.e., if f ∈ Λ(1) ⊂ Λ(2) and g ∈ Λ(2),

Mf g = fg ∈ Λ(2). (6.160)

For now, to avoid convergence problems, restrict M to f ∈ Λ . Thus, if f = P f i , f (1) α∈I0 α α

X Mf = fαMiα. (6.161) α∈I0

Therefore, the analysis can be focused on multiplication by elements of I0.

First, observe that M1 is just the identity operator. Then, to get an expression for a general

Miα for α ∈ I, observe that Eq. (6.8) and

hiα, iβi = δα,β (6.162) lead to σ(α,β) X σ(α,β) hMiαiβ, iγi = (−1) δα∨β,ηhiη, iγi = (−1) δα∨β,γ. (6.163) η∈I

It can be also noted that the adjoint M∗ of M , which, by analogy with the complex case, iα iα can be taken to be the left derivative with respect to iα, is given by

hM∗ i , i i = hi ,M i i = (−1)σ(α,γ)δ . (6.164) iα β γ β iα γ β,α∨γ

Also, because M = M M ...M (6.165) iα ia1 ia2 iaτα and, as a consequence, M∗ = M∗ M∗ ...M∗ , (6.166) iα iaτα iat−1 ia1

153 close attention can be paid to left derivatives with respect to single generators. Thus,

  0, if ak 6= n, ∀ak ∈ {1, ··· , |α|} M∗ i = (6.167) in α  k−1  (−1) ia1ia2 ··· iak−1iak+1 ··· ia|α|, if ∃k; ak = n

Hence, the left derivative constructed here corresponds to the one that is deﬁned in superanalysis [194, 195, 198, 200].

Also, the Berezin integral is a concept widely used in superanalysis and supersymmetry [194, 195, 198, 200]. It coincides with the left derivative and, then, can be deﬁned in terms ∗ of Mi : n Z di f ≡ M∗ f. (6.168) n in

More generally, if |α| < ∞,

Z Z Z ∗ ∗ diαf ≡ dia ··· ia1f = Mi ··· Mi f. (6.169) |α| a|α| a1

Furthermore, if f is generated by ia1, . . . , iaN and iα = ia1 . . . iaN , the Berezin integral

Z diαf = f1,2,...,N (6.170) reduces to Z diαf = hMf 1, iαi, (6.171) which has some resemblance to a residue.

Finally, it is also worth considering the self-adjoint operator

∗ Tf = Mf + Mf . (6.172)

154 ∗ ∗ Then, if f, g ∈ Λ(2) with fB = gB = 0, Mf 1 = fB and Mg 1 = gB. As a result, it holds that

hTf 1,Tg1i = hf + fB, g + gBi = hf, gi. (6.173)

The operator Tf is used later when stochastic processes associated with the FBS space are deﬁned.

6.11 Topological Algebra Associated with Λ(2)

Let S denote the space of Schwartz functions, also known as the space of test functions. Also, let S 0 be its dual, known as the space of tempered distributions. The Gel0fand triple 0 (S , L2(R, dx), S ) plays an important role in analysis [234].

Moreover, Gel0fand triples are also defined in Hida’s white noise space theory [235, 236] and in its noncommutative counterpart [237, 238]. They are used in the solutions of stochastic differential equations and the study of stochastic processes and their derivatives. In this 0 section, Gel fand triples are defined in Λ(2) to solve similar problems. Most of the results presented here find a parallel to the ones presented in Refs. [239, 240].

One of the common reasons for the introduction of Gel0fand triples in the ﬁrst place is the fact that, in a HS, products that are not a law of composition are often considered. As an example, in the white noise space, the Wick product is not a law of composition. Because of that, one embeds the white noise space into a space of stochastic distributions, where the product is a law of composition. This embedding can be made in diﬀerent ways.

Furthermore, even if the convergence of the product is not a problem, the introduction of the space of stochastic distributions might still be necessary. In fact, in the study of stochastic processes, such spaces are necessary for the analysis of their derivatives.

155 Figure 6.2: Schematic representation of a decreasing family of Hilbert spaces with increasing norm, where n is a positive integer.

In this section, the analogous of the space of stochastic distributions is introduced. Before that, a few facts from the complex case as well as from the theory of perfect spaces and strong algebras are brieﬂy reviewed. More information on those spaces can be found in Ref. [241, 242].

Consider a decreasing family of HSs (H , k · k ) with increasing norms, as represented p Hp p∈Z ∞ in Fig. 6.2. The intersection F = ∩p=0Hp is a Fr´echet space, which is a perfect space, meaning that compactness is equivalent to being bounded and compact. This is the fact, in

particular, when, for every p, there exists q > p such that the injection map from Hq into 0 Hp is compact. An important instance is when this injection is nuclear and, then, Hp is 0 ∞ 0 identiﬁed with H−p. The dual F = ∪p=0H−p, the space F and H0 form a Gel fand triple.

The space F0 is endowed with the strong topology, which is deﬁned in terms of the bounded sets of F. Then, F0 is locally convex, and the strong topology coincides with the inductive limit topology — see Ref. [239, Section 3] for a discussion on the subject.

As a consequence, the analysis is, a priori, done in a larger space of distributions, i.e., continuous functionals on the space F0, which is a (non-metrizable) inductive limit of HSs. However, it is in fact done locally in a HS. There are two reasons why this is the case. The

156 ﬁrst one is the fact that the space of distributions, as already mentioned, is the dual of a perfect space. The second reason is the algebraic structure of S−1 and, in particular, the V˚ageinequality, which is presented in Eq. (6.191).

Two of the main results related to such spaces that are relevant here and that illustrate why the analysis is locally done in a HS are:

A set is (weakly or strongly) compact in F0 if and only if it is compact in one of the

spaces H−p in the corresponding norm.

Weak and strong convergence of sequences in a perfect space F0 are equivalent. More- over, a sequence converges (weakly or strongly) if and only if it converges in one of the

spaces H−p in the corresponding norm.

Before introducing deﬁnitions and results in the GA, it should be mentioned that a topological algebra is deﬁned to be an algebra where the product is separately continuous in each variable. It is not a trivial fact, but it can be shown that a strong algebra is a topological algebra — see Ref. [243, IV.26, Theorem 2] and the discussion in Ref. [240, pp. 215-216].

Now, let {cα} be a sequence of positive real numbers such that

X −2d cα < ∞, (6.174) α∈I0 where d ∈ N, and

cαcβ ≤ cγ (6.175) if α ∨ β = γ. Then, for every p ∈ Z, a HS H−p(cα), often simply denoted by H−p, is deﬁned as  

 X X 2 −2p  H−p(cα) = f = fαiα ∈ Λ |fα| cα < ∞ . (6.176) (2)  α∈I0 α∈I0 

157 It should be noticed that H−q(cα) ⊆ H−p(cα) if p ≥ q, as illustrated in Fig. 6.2.

The norm kfk of f ∈ H is deﬁned as H−p −p

X kfk ≡ |f |2c−2p. (6.177) H−p α α α∈I0

For simplicity, hereby, it will be assumed that

cα∨β ≡ cαcβ. (6.178)

In this case, c0 = 1. In fact, if c0 > 1, the inequality in Eq. (6.175) does not hold in general since c0cα > cα = c0∨α. Moreover, if c0 < 1, the condition in Eq. (6.174) is not satisﬁed, as −2d −2d can be observed from the fact that c0 < cα for every α ∈ I, which implies that the sum in Eq. (6.174) diverges.

−2p Now, let f ∈ H−p with cα > 1 if α 6= 0 and c0 = 1. Then, because limp→∞ cα = 0 for every α 6= 0, X 2 −2p lim kfk = lim |fα| cα p→∞ H−p p→∞ α∈I0 X 2 −2p = |fα| lim cα (6.179) p→∞ α∈I0 2 = |f0| .

The next result introduces a V˚age-like inequality, which is the analogous of a result due to V˚age[244] and allows the analysis of stochastic processes to be done locally in a HS. For

158 that, let f ∈ H−q and g ∈ H−p. Hence, using the Cauchy-Schwarz inequality,

X kfgk2 = |(fg) |2c−2p H−p γ γ γ∈I0 2

X X σ(α,β) −2p = (−1) fαg cγ β γ∈I0 α∨β=γ     X  X  −2p ≤  |fα||gβ||fα0||gβ0| cγ γ∈I0  α∨β=γ  α0∨β0=γ  

X  X −p −p −p −p ≤  |f |c |g |c |f 0|c |g 0|c   α α β β α α0 β β0  γ∈I0  α∨β=γ  α0∨β0=γ       X −p −p  X −p −p ≤ |fα|cα |fα0|cα0  |gβ|cβ |gβ0|cβ0  0  0  α,α ∈I0 γ∈I0;∃β,β   α∨β=γ  (6.180) α0∨β0=γ  1/2 1/2

X −p −p  X 2 −2p  X 2 −2p ≤ |f |c |f 0|c  |g | c   |g 0| c  α α α α0  β β   β β0  0    0  α,α ∈I0 γ∈I0;∃β γ∈I0;∃β α∨β=γ α0∨β0=γ  1/2  1/2 X −p −p X 2 −2p X 2 −2p ≤ |fα|cα |fα0|cα0  |gβ| cβ   |gβ0| cβ0  0 0 α,α ∈I0 β∈I0 β ∈I0  2 X ≤ |f |c−p kgk2  α α  H−p α∈I0  2 X ≤ |f |c−q cq−p kgk2  α α α  H−p α∈I0   X −2(p−q) ≤ c kfk2 kgk2 .  α  H−q H−p α∈I0

159 Now, it only remains to be shown that there exist coeﬃcients cα such that

X −2(p−q) cα < ∞. (6.181) α∈I0

For that, let Pn ϕ(a ) cα = e k=1 k (6.182) for every α ∈ I0, where ϕ is such that ϕ(n) form a monotonically increasing sequence for n ∈ N and ϕ(0) = 0. Then, Eqs. (6.174), (6.175) and (6.178) hold. Moreover, if d = p−q > 0,

∞ X 2(q−p) X X −2d Pn ϕ(a ) cα = 1 + e k=1 k . (6.183) α∈I0 n=1 α∈I; |α|=n

Finally, for Eq. (6.181) to hold, choose ϕ such that, for every k ∈ N,

ϕ(ak) ≤ ξak, (6.184) where ξ is a positive real number that will be appropriately chosen. Then,

∞ ∞ X X X 1 e−2dϕ(a1) = e−2dϕ(a1) ≤ e−2dξa1 = . (6.185) e2dξ − 1 α∈I; a1=1 a1=1 τ(α)=1

if it converges. Then, choosing ξ > ln 21/2d, (6.186)

it holds that X e−2dϕ(a1) < 1. (6.187) α∈I; τ(α)=1

160 Also,

    ∞ ∞ 2 X −2d(ϕ(a )+ϕ(a )) X −2dϕ(a ) X −2dϕ(a ) 1 e 1 2 ≤  e 1   e 2  ≤ (6.188) e2dξ − 1 α∈I; a1=1 a2=1 τ(α)=2 and, in general,

 ∞   ∞  X −2d(ϕ(a )+···+ϕ(a )) X −2dϕ(a ) X −2dϕ(a ) e 1 n ≤  e 1  ···  e n  α∈I; a1=1 an=1 τ(α)=n (6.189) 1 n ≤ . e2dξ − 1

Therefore, ∞ n X X 1 1 c−2d ≤ 1 + = 1 + . (6.190) α e2dξ − 1 e2dξ − 2 α∈I0 n=1

The V˚age-like inequality can, then, be written as

kfgk ≤ C kfk kgk , (6.191) H−p p−q H−q H−p

where Cp−q is a positive constant.

With that, it can be also shown that, if f ∈ H−p and g ∈ H−q, with p > q, then

kfgk ≤ C kfk kgk . (6.192) H−p p−q H−p H−q

Now, it is introduced the space

S1 = ∩p∈ZHp (6.193) and its topological dual

S−1 = ∪p∈Z, H−p, (6.194)

161 which are respectively the analogous of the space of test functions and the space of tempered distributions in the complex case.

It is possible to show that the space S−1 endowed with the pointwise product is a strong

algebra. To see that, start by endowing S−1 with the inductive topology. Eq. (6.191) implies

that the product is separately continuous in each H−p, which is equivalent to continuity in

the inductive topology. Furthermore, S−1 inherits the associativity of the pointwise product

from Λ. This space has, then, a BA structure. Therefore, S−1 can be seen as the inductive limit of Banach spaces, which makes it a strong algebra [240]. Observe that the inductive topology is equivalent to the strong topology.

It should be noticed that, if the strong convolution algebra associated to I0 endowed with

the convolution ∨ si considered, a strong algebra that is closely related to S−1 is obtained. Although those two algebras are not isomorphic as a ring, they are isomorphic as locally convex topological vector spaces. As a consequence, it follows from Theorem 3.7 of Ref.

[239] that S−1 is nuclear and, hence, perfect.

Now, let n ∈ N and f ∈ H−p ⊆ H−p−2. Then, it follows that

kfk ≤ kfk . (6.195) H−p−2 H−p

Then, using Eq. (6.191),

kfnk ≤ C kfk fn−1 H−p−2 2 H−p H−p−2

≤ C2 kfk2 fn−2 (6.196) 2 H−p H−p−2 ≤ Cn−1 kfkn . 2 H−p

162 With that, consider the power series

X n F (λ) = αnλ , (6.197) n∈N0 where αn, λ ∈ C, and assume it is convergent absolutely in an open disk with radius R, i.e.,

X n X n |αnλ | = |αn| |λ | < ∞. (6.198) n∈N0 n∈N0

Now, observe that Eq. (6.196) allows the study of the convergence of F (f) in H−p. Speciﬁ- cally, X X kα fnk = |α |2 kfnk n H−p−2 n H−p−2 n∈ n∈ N0 N0 (6.199) X n ≤ α + C−1 |α |2 C kfk . 0 2 n 2 H−p n∈N

Thus, F (f) converges absolutely in H−p−2 if

R C2 kfkH < R ⇒ kfkH < . (6.200) −p −p C2

In conclusion F (f) converges in H−p−2 if

A corollary of this result is that, F (λ) given by Eq. (6.197) converges in S−1 for f ∈ S−1 if the body of f satisﬁes Eq. (6.200). In fact, if f ∈ S−1, there exists an integer q0 such that f ∈ H−q for every q ≥ q0. By the result just presented, the converge of F (f) requires that kfk < R/C , what does not hold in general. However, because of Eq. (6.179), this H−q 2 condition can be rewritten as 2 R |f0| < . (6.201) C2

Finally, it can be shown that f ∈ S−1 is invertible if and only if its body f0 is invertible.

163 To prove it, observe that, if g is the inverse of f and its body is given by g0, then

fg = 1 ⇒ f0g0 = 1 ⇒ f0 6= 0. (6.202)

Conversely, if f0 6= 0, it can be assumed, without loss of generality, that f0 = 1. Then, as just studied, X F (f) = (1 − f)n (6.203) n∈N0 −1 converges if the body of 1 − f is smaller than C2 . However, (1 − f)B = 0. Therefore, −1 g = F (f) ∈ S−1. Furthermore, remembering that F (x) = x in the complex case, g is the inverse of f.

6.12 Stochastic Processes and Their Derivatives

In this section, a close counterpart of the noncommutative white noise space theory [237, 238] is presented for the study of stochastic processes in Λ(2) as well as their derivatives.

First, a brief review of some aspects of stochastic processes is presented. More details can be found in Refs. [233, 237, 245]. The study of Gaussian stochastic processes can be made through the analysis of positive-deﬁnite kernels since there is a one-to-one correspondence between the two notions [246, 247]. In fact, the kernel coincides with the covariance of the stochastic process.

In the framework that is the basis for the model presented here, the processes are associated with positive-deﬁnite kernels of the form

Z (eiut − 1)(e−ius − 1) Kσ(t, s) = 2 dσ(u), (6.204) R u

where σ is absolutely increasing continuous with respect to the Lebesgue measure, dσ(u) =

164 m(u)du, such that the Stieltjes integral converges, i.e.,

Z m(u)du 2 < ∞. (6.205) R u + 1

The reason for such choice is the fact that integrals of the type given by Eq. (6.204) correspond to correlation functions of zero-mean Gaussian processes with stationary increments [248, 249]. An important example of such processes is the fractional Brownian motion, for 1−2H which dσ(u) = |u| du, where 1 < H < 2. In this case, the correlation function Kσ(t, s) becomes 2H 2H 2H K(t, s) = γH |t| + |s| − |t − s| , (6.206)

where γH depends only on H. If H 6= 1/2,

cos(πH)Γ(2 − 2H) γ = , (6.207) H (1 − 2H)H

where Γ is the Euler’s Gamma function. Moreover, by continuity, γ1/2 = π.

Also, it is introduced the following operator Sm in L2(R):

p S[mf(u) = m(u)fb(u), (6.208)

where fb is the Fourier transform of f. Note that Sm is, in general, unbounded. Its domain is Z 2 dom Sm = f ∈ L2(R) m(u)|fb(u)| du < ∞ , (6.209) R

which contains 1[0,t]. Deﬁning

fm(t) = Sm1[0,t] (6.210)

165 and using Plancherel’s equality, which assures that

1 D E hf (t), f (s)i = f (t), f (s) , (6.211) m m L2(R) bm bm 2π L2(R)

it holds that

1 D E hf (t), f (s)i = pm(u)1 , pm(u)1 m m L2(R) b[0,t] b[0,s] 2π L2(R) * + 1 e−iut − 1 e−ius − 1 = m(u) , 2π u u (6.212) L2(R) 1 Z (eiut − 1)(e−ius − 1) = 2 m(u)du. 2π R u

For the stochastic processes of interest here, a random variable is associated with the functions fm(t). This is done with the introduction of the creation operator `h, with h ∈ L2(R), deﬁned by

`h(f) ≡ h ⊗ f, f ∈ Γ(L2(R)), (6.213)

∗ where Γ(L2(R)) denotes the FBS space associated with L2(R). Finally, letting Th = `h +`h, a random variable Xm(t) is deﬁned as

X (t) ≡ T . (6.214) m fm(t)

Observe that the expected value of a random variable Xm(t) can be deﬁned by

D E E(Xm(t)) = Ω,T (Ω) , (6.215) fm(t) Γ where Ω is the vacuum state of Γ. Moreover, as expected,

D E E(Xm(t)Xm(s)) = T (Ω),T (Ω) = hfm(t), fm(s)i = Kσ(t, s), (6.216) fm(t) fm(s) Γ L2(R)

166 where Kσ(t, s) is given by Eq. (6.204) with dσ(u) = m(u)du, as already discussed.

Finally, for the processes of interest here, it is assumed that m satisﬁes

  K|u|−b |u| ≤ 1, m(u) ≤ (6.217) 2N  K|u| |u| > 1, where b < 2, N ∈ N0, and K is a positive real constant. Then, letting ξn denote the Hermite functions, N+1 |Smξn(t)| ≤ D1n 2 + D2, (6.218) and N+2 |Smξn(t) − Smξn(s)| ≤ |t − s| D3n 2 + D4 , (6.219) where D1, D2, D3, and D4 are non-negative functions independent of n. The proof for this result are the Proposition 3.7 and the Lemma 3.8 of Ref. [250].

Now, in the GA, the operator Tf in Eq. (6.214) is replaced by the operator deﬁned in Eq. (6.172) with X D E fm(t) = Sm1[0,t], ξn in L2(R) n∈N X D E = 1[0,t],Smξn in (6.220) L2(R) n∈N X Z t = (Smξn)(u)du in. 0 n∈N where ξn denotes the Hermite functions. Hence,

X Z t Xm(t) = (Smξn)(u)du Tin. (6.221) 0 n∈N

Moreover, the expected value function E can be deﬁned in a similar manner as deﬁned in

167 the well-known cases, i.e.,

E(Xm(t)) = h1,Xm(t)1i , (6.222) Λ(2) where 1 is the vacuum state in the FBS space introduced in Section 6.10. Note that

E(Xm(t)) = 0.

Furthermore, using Eq. (6.173), it can be observed that the covariance, which gives the kernel K(t, s), satisﬁes

E(Xm(t)Xm(s)) = K(t, s) = hXm(t)1,Xm(s)1i = hfm(t), fm(s)i . (6.223) Λ(2) Λ(2)

Hence, Z t Z s X 0 0 K(t, s) = (Smξn)(u)du (Smξn)(u )du , (6.224) 0 0 n∈N which is equivalent to the kernel deﬁned in Eq. (6.204).

The goal here is to show that the framework just introduced allows the study of the derivatives of stochastic variables as continuous functions. However, a series a results has to be

presented before that. To start, observe that the operator Mf is bounded from Hp into H−p for every f ∈ S with kfk < ∞. In fact, if g ∈ H , then, using Eq. (6.192), it holds −1 H−p p that

M g = kfgk ≤ C2p kfk kgk . (6.225) f H−p H−p H−p Hp

∗ Also, for every f ∈ H−q, the operator Mf is bounded from Hp into H−p, where q < p.

168 Again, letting g ∈ Hp, this result can be seen from

2 2 2 ∗ ∗ X X σ(α,γ) 2p Mf g ≤ Mf g = (−1) fαgβ cγ H−p Hp γ∈I0 α∨γ=β  2 X X 2p ≤  |fα||gβ| cγ γ∈I0 α∨γ=β X −p 0 −p X p 0 p ≤ |fα|cα |fα|cα0 |gβ|cβ|gβ|cβ0 0 0 α,α ∈I0 γ∈I0;∃β,β α∨γ=β (6.226) α0∨γ=β0 X ≤ |f |c−p|f0 |c−p kgk2 α α α α0 Hp 0 α,α ∈I0  2 X −(p−q) ≤ |f |c−qc kgk2  α α α  Hp α∈I0 ≤ C2 kfk2 kgk2 . p−q H−q Hp

It should be also noted that, for every f ∈ S−1, it is a consequence of the last two results ∗ presented that the operator Tf = Mf + Mf is bounded from S1 into S−1. Moreover, there exist p > q such that

kTf gk−p ≤ 2C1kfk−qkgkp. (6.227)

In fact, observe that, if f ∈ H−q and g∈Hp, it holds that

∗ kTf gk−p ≤ kMf gk−p + kMf gk−p

≤ Cp−qkfk−qkgk−p + Cp−qkfk−qkgkp (6.228) ≤ 2Cp−qkfk−qkgkp

≤ 2C1kfk−qkgkp, where Eq. (6.191) was used.

169 Now, it can be ﬁnally shown that the derivatives of the stochastic variables considered here

are constant in S−1. This fact is stated as follows: Let m be a positive measurable function

that satisﬁes Eqs. (6.217) and (6.205) with dσ(t) = m(t)dt. Then, for every g ∈ S1, the function t 7→ Xm(t)f is strongly continuous in S−1 and there exists a continuous operator

Wm(t) from S1 into S−1 such that

d X (t)g = W (t)g. (6.229) dt m m

Finally, the function t 7→ Wm(t)g is continuous from [a, b] into S−1.

To prove this result, observe that, since

X m fm(t) = fn (t)in, (6.230) n∈N with Z t m fn = Smξn(u)du, (6.231) 0

Eq. (6.218) leads to

2 d X 2 −2p X N+1 2 −2p fm(t) = |Smξn(t)| cn ≤ (D1n 2 + D2) cn . (6.232) dt H −p n∈N n∈N

Then, for every suitable choice of coeﬃcients cn, there exists a positive integer p0 such that,

for every p ≥ p0, dfm(t)/dt ∈ H−p. Moreover, for s = t + h, where h 6= 0 is a real number,

there exists p1 such that,

X N+2 2 −2p Kp = (D3n 2 + D4) cn < ∞, ∀p ≥ p1 (6.233) n∈N

170 and, using Eq. (6.219), the H−p norm of the diﬀerence of derivatives satisﬁes

X 2 −2p 2 |Smξn(s) − Smξn(t)| cn ≤ Kp1|h| . (6.234) n∈N

Therefore, W = T is a continuous operator from S into S . To see that Eq. m dfm/dt 1 −1

(6.229) holds, observe that, for a real number h 6= 0 and g ∈ Hp,

R t+h Xm(t + h) − Xm(t) X (Smξ(u) − Smξ(t))du − W (t) g = t M g = X g, h m h in ∆(t,h) n∈N (6.235) with R t+h X (Smξ(u) − Smξ(t))du ∆(t, h) = t i . (6.236) h n n∈N

Thus, there exists p > q ≥ p1 such that, using Eq. (6.227),

X g ≤ 2C k∆(t, h)k kgk ≤ 2C K kgk |h|2. (6.237) ∆(t,h) 1 H−q Hp 1 p1 Hp H−p

This shows that W (t) = T (t) ≡ dX (t)/dt is a continuous operator from S into m dfm/dt m 1

S−1. Finally, since a sequence converges in S−1, i.e., a perfect space, if and only if it converges in one of the spaces H−p (as stated earlier in the section), it can be said that the function t 7→ Wm(t)g is continuous.

Now that the treatment of derivatives is formalized, the next step concerns, for instance, the development of the counterpart of Ito or Malliavin stochastic calculus. The ﬁrst step in this direction is the introduction of stochastic integrals. This is done in the ﬁnal result of this dissertation. The stochastic calculus itself will be developed in future works.

Let t 7→ Y (t), with t ∈ [a, b], be a continuous S−1-valued function in the strong topology of

S−1. If g ∈ S1, there exists a positive integer p, which depends on g, such that the Pettis

171 integral Z b Y (t)Wm(t)g dt (6.238) a can be computed as a limit of Riemann sums and converges in H−p.

The proof is similar to the one presented in Ref. [250]. Since the function t 7→ Wm(t)g is continuous from [a, b] to S−1 and the product is jointly continuous in S−1, the map t 7→ Y (t)Wm(t)g is also continuous and its image is, therefore, compact in S−1. Now, recall that a set is (weakly or strongly) compact in S−1, i.e., a perfect space, if and only if it is

compact in one of the spaces H−p (as stated earlier in the section). Then, there exists p

such that the image of t 7→ Y (t)Wm(t)g is in H−p. This function is also continuous with

respect to the topology of H−p. Then the integral in Eq. (6.238) can be computed in H−p as a Riemann sum.

To conclude, it should be noticed that, as already discussed, the random variables introduced here have the same expected value and the same covariance of the variables associated with free stochastic processes [233]. Despite that, observe that the product in the integral in

Eq. (6.238) is, ultimately, the product of iα’s, which contrasts with the product of Hermite functions in free stochastic processes. This shows that the processes induced by them are drastically diﬀerent and should be further explored.

6.13 Discussion

In this chapter, analysis on two diﬀerent closures of the GA were studied, namely the closure

with respect to the 1-norm, Λ(1), and the closure with respect to the 2-norm, Λ(2).

In the study of Λ(1), the notion of positivity and the CK product for power series of a variable

in Λ(1) were introduced. Such ideas were used to study the counterpart of classical problems, such as the one-step extension problem for Toeplitz matrices and the Wiener algebra, and

172 to begin the development of Schur analysis in this setting.

The extension of these notions from a single complex variable to more general settings, like several complex variables, upper-triangular operators, quaternionic analysis, and bi-complex numbers, has been a source of new problems and methods [86–91]. Each of these settings has some interpretation in terms of signal processing theory and linear systems. For example, time-varying systems correspond to upper-triangular operators, and systems indexed by N several indices correspond to function theory in the unit ball of C , also known as the unit polydisk. Furthermore, in all those settings, there exists a natural counterpart of the Hardy space. In the case of the upper-triangular operators, it is the space of Hilbert-Schmidt upper-triangular operators. As to the case of function theory in the unit ball, it is the Drury-Arveson space [84], which diﬀers from the classical Hardy space when the dimension N is greater than 1. In the setting introduced here, the counterpart of the Hardy space is a Wiener-type algebra.

Multiple research directions emerge from the results that were presented in this chapter. For instance, the problem of the one-step extension of Toeplitz matrices in complex analysis, as mentioned in Section 6.3, is associated with stochastic processes and, in particular, with the Yule-Walker equations. Then, it can be questioned if there is a version of those equations in Λ(1). If there is, are their solution also connected in the same way to the problem of extension of Toeplitz matrices? Moreover, what are the stochastic processes that arise from this problem? Do they have any similarity with the class of processes introduced in Section 6.12?

Furthermore, only a few foundational tools from functional analysis were presented here. There is still a lot to be uncovered — from results that can be translated from other settings to results that fail to hold in Λ(1).

In the last sections of the chapter, counterparts of notions from the theory of complex analysis

173 and stochastic processes were developed in Λ(2). In particular, given the deﬁnitions already presented, the Gel0fand triple

S1 ⊂ Λ(2) ⊂ S−1 (6.239) was considered. It should be mentioned that the space S−1 has an algebra structure of the type that was ﬁrst introduced by Kondratiev in the setting of Hida’s white noise space theory [235], and studied in a more generalized framework in Ref. [240]. There are many parallels (and diﬀerences) between the results presented here and other works where the complex numbers were replaced by the commutative algebra of Kondratiev stochastic distributions [235, 251–254]. Here, the main objective was the development of a framework that allows the introduction and analysis of stochastic processes and their derivatives.

The stochastic processes that were used as a basis for the one presented here have the concept of freeness — in opposition to independence — associated with their random variables. Also, the distributions associated with them are semi-circles — in opposition to Gaussians. A possible research direction is, then, the investigation of the questions: What is the independence-like concept associated with the random variables deﬁned in this chapter? What are the distributions associated with the stochastic processes generated by them? Moreover, as already mentioned in the previous section, the counterpart of Ito and Malliavin stochastic calculus in the present setting has still to be developed.

Furthermore, another interesting direction is to look into generalizations of the class of processes that was considered here. In Section 6.12, the special cases for which dσ(t) = m(t)dt were considered. For a general σ, the operator Sm deﬁned by Eq. (6.208) cannot be introduced. However, it is possible to prove that there exists a continuous positive operator A from the Schwartz space S into its dual S0 such that

Z 2 |fb(u)| dσ(u) = hAf, fiS0,S. (6.240) R

174 Since S is nuclear, the operator A can be factorized via a HS [255]. An explicit construction ∗ of A in the form A = QσQσ, where Qσ is continuous from S into L2(R), is given in Ref. [256]. It can, then, be investigated whether these results lead to a generalization of the processes presented here.

Finally, due to the importance of Grassmann numbers in quantum ﬁeld theory and the physical motivation of the complex counterpart of what was introduced here, it can be asked what the applications (if there is any) in real-life problems of the work displayed in this chapter are.

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